Attempting to resolve the SSB/auto pilot issue can be more trouble than it's worth. I say just turn the AP off while transmitting.

On our vessel since the Admiral is the one that works with the SSB most of the time (emails, nets, net control, etc.) I just retreat to the cockpit, disengage the auto pilot and,, if were under engine I shut it off and just ghost along in some reasonable direction until she's done (if there's no wind at all I take the engine down to idle.) She doesn't want me in her way anyway and if the engine is running there's quite often too much noise for her to decipher some of the voices (even with the Sony headset on). I might as well have something to do. If it's you that handles the SSB then send some one else to the cockpit to hand steer.

For us that approach works fine and the entire SSB/auto pilot thing is a non issue.

Since the reason the auto pilot goes erratic is the transmitting power you might lower the transmit watts down to 50 instead of 100 or more (I assume you're probably transmitting on high power as most people do it even though it's not necessary). Lovering the transmitting watts may resolve the problem.

I say hand steer and be done with it. Spend your time resolving more urgent issues.