"Brit" is generally fine.
If you want to impress a native while in Albion, brush up on your geography and general knowledge of Britain. I think we're easily impressed by U.S. Americans especially who make the effort. But of course, one notable error like thinking Cornwall's a city will mean you're back to square one. Perhaps learn a dozen or more cities in the British Isles if you don't know that many already, and similarly with our counties.
Another source of respect would be correct pronunciation and stressing of place names and I'll throw down some examples because it's 2am and this is what I do:
Birmingham = Burr-ming-UM (not Burr-ming-HAM/-HAYUM)
This same rule applies for most "-ham" towns. Durham = DUH-rum.
Scarborough = SCAR-bur-UH (not Scar-BRO)
This same rule applies for most "-borough" towns, including cases like Middlesbrough and Edinburgh.
Cheshire = CHESH-uh (not Che-SHY-uh)
This same rule applies for most "-shires" despite "Shire" being pronounced "SHY-uh", although examples like CHESH-eer, SHROP-sheer and WUSS-ter-sheer are also used.
Leicester = LESS-ter
This same rule applies for examples like Worcester (WUSS-ter) and Gloucester (GLOSS-ter).
Plymouth = PLIH-muth
This same rule applies for examples like Portsmouth (PORTS-muth) and Cockermouth (Cock-UH-muth).
Then there are other strange variables, eg. Southwark = SUTH-uk, Keswick = KEZ-ick, Norwich = NOH-rich, Salisbury = SOLS-bury.
I've forgotten what I was attempting to achieve with this or why, but that should make anyone well-prepared and leave us natives appreciative.
Middlesbrough is Bruh NOT buruh. That is how to offend a teessider.