Is the rest of the tank pretty pristine or does it have other issues as well like maybe GHA, cyano, Dinos, vermetid snails or any other issue?
If you have some of these issues as well then i understand a clean re-boot.
If the rest of the corals are grown out, other inhabitants happy & tank is looking/doing fine then id can the tang or house it & get the file fish & try to figure out what happened to the shrimps rather then having to clean & re-cycle everything from day one again.
Took me 1yr. to get completely cycled, id only do that again if i was upgrading larger or as a absolute last resort after trying everything else if i had multiple bad infestations going on.
I have vermetids ATM so feel your pain but so far they are the small ones and not yet bothering anything except me so far.
Without pics or more info this is what id do at this point before jumping ship & starting over if just 1 tang is the problem stopping you.