Miss Pettigrew Lives For A Day is exactly the sort of screwball comedy audiences back then would have loved.
I agree with Amadeus, but I’d also nominate Nannerl, Mozart’s Sister. It would have appealed to audiences who back then were used to strong female characters. (I saw the movie twice with subtitles, but this short trailer showing the look of the film is without subtitles)
No thrillers. No sci-fi. No gross-out summer flicks. No pretend-sophisticated sex comedies. No action blow-em-ups. No Tarantino. No irony of any kind. What’s left?
I think the best candidates would be somewhat shmalzy and mystical. Like Meet Joe Black.
Kitt Kittredge: An American Girl
Bet there’d be a lot of head-scratchin’ about that one.
Actually, the movie does have the scene where the wreck and sinking are recreated, right before old Rose begins her story.
But then we’re left with explaining exactly what the characters were looking at - “If you look at my invention, the Ray-Screen, which is calibrated to show images, I have devised a series of animated pictures to recreate the sinking of the Titanic…”
How about Chicago, then?
“My name . . . is Gladiator.”
Pretty gory for 1930. There was a time when even showing blood spattering from a bullet wound was controversial.
If it helps, the OP said “1930s”.
Blood spatters didn’t really get shown until the 60s. It was controversial even then…a lot of people were upset by the “graphic violence” in Bonnie and Clyde and The Wild Bunch.
The actual story itself, definitely- but I don’t think it passes the “How the hell did they actually make that?” technology test; even when it was released it looked like a computer game with live actors in it. (Great film, though!)
How about the recent Call Of Cthulhu film? It’s a modern movie done in the style of a 1920s silent film (and it’s awesome). The time period is OK (late 1920s) the special effects aren’t anything that wasn’t available at the time, there’s nothing in there that would be objectionable to an audience in the 1930s. And silent films were still being made (for artistic reasons) until at least 1936, so no issues there either.
The other nomination that springs to mind is the Brendan Fraser version of The Mummy, which starts in 1926 and most of the action takes place in the early 1930s (Twenty Minutes Into The Future by their standards). I don’t recall it having any scenes of graphic violence that couldn’t be fixed with some judicial editing, and the CGI special effects could conceivably be explained away with “Really good models” and “Talented artists painting the effects onto the cels”.
I’d send something back like White Chicks, Black Dicks XII or something, just to mess with people.
I’d probably not do well…
If I can go just a few years before the time limit set by the OP then my choice would be The Princess Bride. This movie actually didn’t do well at the box office in 1987, it became popular on home video, but I think audiences in 1930 would be quicker to recognize its charms. There’s a dashing leading man, a beautiful leading lady, great supporting characters, and clever dialogue that doesn’t depend on familiarity with late 20th century pop culture. The basic plot, setting, and action scenes are similar to the type of swashbuckling romances already popular in the late 1920s. The effects were not exactly cutting-edge even in 1987, so no problems there. I’m not sure if the expression “son of a bitch” or the reference to Buttercup’s “perfect breasts” would be considered acceptable in 1930, but if these were dubbed over (“son of a cur” and “perfect bodies”?) then I don’t think anything else would be considered objectionable.
As with Titanic the framing device is potentially problematic, but the shot of Fred Savage’s video game could be cut and the brief references to video games by him and Peter Falk could be dubbed over. I’m not sure how how much the appearance of Savage’s bedroom would throw 1930s viewers out of the movie, though. It’s only seen briefly, but the fact that it’s cluttered with (from a 1930s perspective) really weird looking stuff might be distracting. If we were allowed to re-shoot or do some crafty CGI work/redubbing to have these scenes set in 1930 then that would eliminate anachronism without significantly changing the movie. It would also actually be closer to the original book. The boy in the novel is a fictionalized version of William Goldman, and that part of the story is set in the late '30s or early '40s when Goldman was young.
I agree, Beauty and the Beast would be a great choice. It would be the most impressive animation audiences in 1930 had ever seen, but animation had been around for a while by then and Wikipedia tells me there had even already been a German animated feature that used color tinting, so it probably wouldn’t provoke the “I honestly don’t understand how that film could have been made” reaction the OP warns against.
I wouldn’t have thought of Moulin Rouge!, but it might work. The movie actually makes significant use of CGI – I believe all the “exterior” locations are CGI – but given what people like Fritz Lang had already managed to pull off with the use of cleverly constructed sets, matte paintings, models, etc., 1930s viewers could probably accept all the visual effects in Moulin Rouge! as more of the same. Despite being rated PG-13 I don’t think there’s much in the movie that would shock 1930s audiences, although the scene where Nicole Kidman pretends to be VERY excited by Ewan McGregor’s poetry reading would probably have to be edited. (And arguably should have been edited anyway, just because it wasn’t that funny.)
The basic story borrows heavily from works that predated 1930: La Dame aux camélias/Camille, La Traviata, and La Boheme. There had already been several silent film versions of Camille, but a 1930 release date for Moulin Rouge would beat the Greta Garbo talkie version by six years. Although plenty of late-20th century pop music is featured in the movie it can be enjoyed even by those who don’t already know the songs. Actually, since this was an element that annoyed some 2001 viewers it might play even better with mainstream audiences if they don’t know the songs already. This does raise the possibility of a Back to the Future-esque scenario where later musicians are inspired by bits of songs from an old musical when they actually wrote those songs in the original timeline, but the OP didn’t ask us to consider that sort of thing. 
There were reports right after the sinking where survivors claimed to have seen the ship split before it sank. White Star did it’s best to discredit them (it being considered a crack pot theory until the wreck was discovered. I think Titanic could still make a coherant narritive without the framing scenes set in “present day”, but some things would be very odd. Like Old Rose’s narration; the audience would be able to understand it’s Rose in the future without seeing her, but would wonder why she sounds so old and why her words suggest so much time has taken place when it was only 20+ yrs ago and she’s still in her 30s. Also why is that necklace getting so much attention? And the ending with her getting off the Carpathia under Jack’s name would seem abrupt, but it might be possible to work some of her photo’s in somehow.
You’d also want advances any references to dates in the film by 20 yrs so that it takes place in 1959. The audiance would have an easier time with the idea that it simply takes place in the future instead of an alternate version of the present.
I’m thinking a smartly written romance - “As Good as it Gets”… however there might be a slight problem with Greg Kinnear’s character and Jack’s potty mouth…
“Cinderella Man” would really enthrall people. It would be familiar, extremely topical for the time - almost painfully so, and boxing was one of the most popular sports in the world at the time…
Groundhog Day
Wait, that sounded like a time-travel joke & was not meant to be. I just like Groundhog Day. But the television technology jokes would be the reverse of dated, so never mind.
Um, Joe Vs. the Volcano would really work for that era I think.
Crap, you said, 20 years.
Um, I have no idea.
The plot of Harry Potter would translate pretty well to the 1930s, IMO. But the movies would never get past the “How’d they do THAT?!??” barrier.
A River Runs Through It would be set in the very recent past. You would have to edit out the brief nudity and possibly some language.
I think Back To The Future would be the funniest choice. You’d have people in the '30s looking at the '50s in THEIR future and wondering why the hell anyone from the '80s would want to go back there.
I know it doesn’t exactly fit the rules of the OP, but think of the madness you could cause! I mean, look at the gorgeous cars!
Year One
10,000 BC
Silverado
Tombstone
310 to Yuma
(actually, pretty much every modern western)