How do I pronounce "can't" with a British accent without it rhyming with "punt" or "paint"?
There are a great many pronunciations of can't in British English. I assume you mean a standard RP /kɑːnt/. In that case you need a vowel which you don't have in American English, the long back a. Fortunately, you have a short back a in words like palm and lot (in Standard American English), the open back unrounded vowel. You need to say the vowel in the middle of those words but make it longer.
It's a pure vowel, so first try saying it in a long, extended sound to make sure you understand the articulation in your mouth. In Britain this is the sound our dentists make us say so you get an impression of the shape that would be to allow a dentist to gain access. [Yes, British dentists do exist, ;-)].
When you can do that, try saying the vowel about twice as long as you normally would, but put the (uncomplicated) consonants at the start and end.
When you can reliably join those sounds together, listen to someone speaking the word and modify the length of the vowel accordingly.
Try to avoid regional or Scottish or Welsh accents (unless they're you're target) or extremely posh people, as this is a vowel sound with a great variety in England: it is one of the most important regional differences in England.
The link below sounds reasonably standard modern British English to me.