Go with your gut instinct, not with their determination to follow 'proceedure' (if I hadn't gone with my instinct, Polly would have been pts unnecessarily). A well-informed, loving owner knows their cat better than any vet. If you don't think the tests are necessary, don't have them done.
I wouldn't blame your own vet for 'not sending the test results' - he probably has, but this specialist has mislaid them, or is ignoring them. They don't seem to realise that 3000 is a lot of money, even when you're insured. Maybe because they are specialists, they do expensive tests all the time, and think nothing of it.
It does sound like Mango's problem is with his nose, especially as you say he's had the problem from birth - that suggests a congenital defect, rather than something like cancer, which he wouldn't have been born with.
Don't be intimidated by these people, it's YOUR cat and YOUR insurance, and ultimately YOUR decision, not theirs. Tell them, then tell them again, (don't be diverted by their arguments or attitude) just tell them again what YOU want them to do. (I did an assertiveness course years ago, and this is one of the tricks). Stay calm, but don't let them digress, just say "no, I would like you to...." until they get the message, regardless of how many "no thanks, I want this done ...." you have to say.
Don't blame your own vet, it seems like any screw-up has happened with these so-called specialists.
Good luck to you and Mango.