How to match harmonica impedance to a tube amp to eliminate feedback
You seemingly want certain distorted sound and reach it in the same way as rock guitarists - by cranking the gain up until the amp distorts enough. Unfortunately the feedback caused by high gain with acoustic mic easily makes the system to squeal. The hands around the mic +using some high midrange peaking mic (possibly designed originally to increase the clarity of human voice) make things worse. Things become even worse if high onstage sound level force the harmonica player to turn his amp or monitor speaker louder just to be able to hear what he plays.
Modern DSP has given automatic feedback suppressors which really can detect starting squeal and equalize the peaking frequency gain down to a safe level. A little older tricks are

to have a normal parametric or terse equalizer which attenuates the most easily squealing frequency range

to use a noise gate which automatically shuts the mic totally in silent places and

a few Hz pitch shift which makes feedback to have a little wrong frequency for fast buildup


I recommend lower onstage sound levels. A loud amp for right sound can be enclosed to isolation box or use attenuated speaker. Simply moving it further also reduces feedback, but something extra may be needed to let the player hear himself. In-ear monitor can keep excessive noise out and make the wanted sound well hearable for the player. I have met singers who have purchased a personal earplug monitor system which actually outputs only their own voice and attenuate everything else purely mechanically (=very small holes for external sound)