To me “Brit” is like “Yank”. It’s what we call each other for short. No biggie.
“Limey” might be a bit of a slur, and I have to wonder if Americans in the South can compartmentalize “Yank” as being distinct from “Yankee”. I suspect they could, if being referred as that by a Brit. I can’t imagine a Southerner using the term “Yank” to refer to Northerners. It just doesn’t have the same feel to me, as an American. But then, I’m not from the south-- I’m a Yankee. But not a Yankees fan!
Which, even though I already knew it, made me giggle quite hard during an episode of Misfits when one character asked another if she shaved her fanny.
I can’t honestly imagine many people getting in the least bit upset at “Limey”. As a nation we do like to give unflattering nicknames to other nationalities, but it is usually done light-heartedly and with some affection. Why would we get upset in return?
If you need an alias and you want to remain unobtrusive, don’t call yourself Randy. Also, for some people tea is a drink but for others it’s also a mealtime. Hence, “You’re interrupting my tea.” likely doesn’t mean they just had to put down a cup to answer the phone.
Perhaps he’s trying to get a deduction on his taxes.
You go call a british pakistani a paki and see how your “a pakistani told me it’s fine.” defense goes down. Please get video as well, I need a laugh.
I’ve actually come across English-Pakistani kids calling each other ‘Paki’ (as in ‘you’re not a real Paki anyway, your Dad’s Indian’, ‘Am so a Paki!’), but I wouldn’t try it if you’re not of Pakistani background.
Personally, ‘Brit’ grates somethin’ awful on me. You can’t go far wrong with plain British. Oh and have fun with our more… interesting accents. 
Well, that is like how African Americans can call each other “nigger”, but the rest of us certainly can’t call them that. “Paki” is much the same in Britain.
Although I will use “Brit” in some circumstances, and it is certainly not deeply insulting, I think it might still raise my hackles a little bit if an American telemarketer addressed me as one.
One point for the OP: Do not mention or imply to a British person (or to any European, come to that) that America “saved their asses” (or anything similar) during the two world wars. (It is perfectly fine to note that the countries have long been allies, in the unlikely event that it should happen to come up, or even that Britain and America have a “special relationship”.)
I suspect bringing up wars in general is considered a no-no in telemarketing.
I suspect the problem won’t come up, because most telemarketing joints will heave you out the door if you vary a syllable from their scripts.
We get a surprising amount of leeway with our scripts, meaning “a little,” but I need to be prepared to make small talk because I’m so doggone charming on the phone that people love to talk to me. That is, of course, if I remember to switch on my salesman personality. Otherwise they get the same annoying and unpleasant dropzone you people have been stuck with for twelve years.
I mentioned it once, but I think I got away with it all right.
I do not expect to actually call anybody a Brit, much less a Limey, especially because the latter flows right into “bastard.”
I’m a Brit, and a Limey and not offended by either term.
I’ve had Brit used to me by Republican Irish intending it to be an insult, but haven’t been offended by it as I’m not ashamed of my nationality or culture.
Whitey doesn’t offend me either as I am indeed white, and more then happy with who I am .
I’m an Australian telefundraiser and last year did a stint of ringing England for an international charity.
One thing I found (and it doesn’t answer your question in the least!!), but a LOT of people mentioned that they were OAP’s. Literally…‘I’m an OAP’, and I had no idea what on earth they were saying.
On further investigation, OAP stands for Old Age Pensioner. In Australia they would have just said, “I’m on a pension luvvie”, but OAP had me stumped.
Good luck with the dialect differences dropzone. And being a telemarketer really isn’t the end of the world. 
Don’t mention the war! 
I mentioned it once, but I think I got away with it !
I was the only American in a UK band for awhile; I got the gig because they lost a guitarist and the bassist in Austin, Tx.
It didn’t take me long to pick up on certain phrases and words, like “The Loo”, “Oh, Bollocks”, “Wanker” and “Pissed”-as-in-“drunk”.
I’ve also started asking, “Well, that’s a bit stupid, isn’t it?” all by simply being immersed in Brit Speak for 2 months back in 05.
“You’d all be speaking German if it wasn’t for us…”
Really insulting to Germans.
Especially when a German answers the phone in Auchtertool, Scotland.
But the same could be said for French, Dutch, and any other language you could think of.
We’ve been fighting each other for several hundred years
We don’t actually care .
We’re the best .
The Krauts think that they’re the best
The Dutch also
The Scandinavians
We’re the fucking best …
Bar none.
But we don’t honestly care.
Americans do,
Arabs do .
But we don’t care.
THAT is useful knowledge!
I have no problem with being a telemarketer, though the job is perceived negatively by some Dopers. It’s not a continuation of my career but it’s honest work and the company is better than some I’ve worked for. I may hate talking on the phone and can’t imagine how people who are not getting paid to do it spend hours gabbing away, but that’s partly because of the times I HAVE been paid to talk on the phone. It makes doing it for free seem odd. I suppose many prostitutes don’t much care for sex when they are off the clock, too.