Hello,
There are still quite a few old timer vets who may be in your area and willing to help. Call everyone around you. Try to avoid the corporately owned clinics, and ask for a reference from everyone at the local shelters, rescues and via every social media outlet you use (facebook, nextdoor, etc). In many cases if these are older cats then these are either masses due to chronic infection and inflammation, or a neoplastic mass.
If this is a younger cat it may be an oronasalpharyngeal polyp.
Keep asking for affordable options and keep advocating for your cat,
Hello,
I apologize for the delay. Somehow I missed this one.
Call everyone and see if you can find someone who will anesthetize and look for a polyp behind the soft palate. Call all of the rescues, shelters and private veterinary clinics. Put out a plea on social media. Use all platforms to ask for help.
Keep me posted.