Never heard the term “Christmas tiger” (or at least never noticed it, because I have seen MSGtW) but the “bobbing head tiger” is an item from traditional Japanese folk-art (collectively known as mingei) that made it to the US. Originally called hariko-no-tora. See this entry. (I have a deadtree copy of this, it is a small, “field guide” sized book. Nice to own if you are interested in the subject.)
Looking into the subject because of this thread, I found that bobbing head tigers from Japan became something of a fad in America in the sixties. Which is not too surprising; the post-war American occupation saw a lot of cheap Japanese goods being exported to America.
But this doesn’t answer the OP’s question. Mr. Smith Goes to Washington was released in 1939, which indicates there was pre-war knowledge of these items as well.
Just a wild guess. But maybe it refers to how people will get worked up about a thing when all around them do. Jumping on the band wagon. A Christmas Tiger might refer to a change in some peoples behavior around Christmas. Whereas the rest of the year they might be more of a scrooge.
In 1938, ESSO/EXXON issued as it’s Christmas premium (gift with purchase - common well into the 1980s) a tiger whose head bobbled when you put it on the dash (front or rear) of your car. Similar to the ones people used to have in the 1970s - except those were dogs. I recall on Chico and the Man, Freddie Prinze says something about the doggie ones (I was only about 8 or 9 at the time and didn’t really follow). This site talks about bobble heads in general but doesn’t actually talk about Esso/Exxon’s Tiger (it’s mascot): History & Anatomy of Bobble Heads - Bobble Head Shop – PopCultureSpot.com - Shop for Bobble Heads, Novelties, Stickers — 25th Anniversary!
This question was one I had for many years myself - until I started seeing some of the old premium items Esso/Exxon gave away in flea markets and now in “antique” shops. If anyone ever sees the Tiger, I hope they’ll post a photo of it.