This would be a very simple job for a drain cleaner if you’ve exhausted all your options. Total job should be about $50. Trap may have to be replaced no matter who removes it due to age. Check the pop-up stopper as a last resort, often hair and soap build up right there. Lane.
Removing the trap will only give access to the problem. The stoppage is not in the trap, unless you dropped a pencil or similar down it. Traps are designed, by code, not to catch hair, etc., so you would still need a snake after you removed the trap.