I did a google search on your zetig amplifier, it has a SSB delay so it will work on SSB.
Common CB myth: "my amp has a SSB delay, so it will work on SSB". No. The SSB delay on most of these amps is simply a capacitor that is switched in with the relay that prevents the relay from chattering because there is no carrier to keep the relay engaged between voice peaks. That alone does not make it suitable for SSB. For SSB, it's important that linearity is maintained which is nearly impossible with a class c amplfier inline.
I compared the schematic with one I found on the web for class c mosfet rf amplifier and the relevant sections look pretty much the same. I also looked at a schematic of an AB biased mosfet amp. Please understand that I'm not very versed with MOSFET PA design so I could be very wrong, but I can't see how that amp you have is much different than the class C design I found. It is possible that you have other issues like drawing too more current from a battery than it is capable of, but I doubt it. FM and AM modes typically draw more current than SSB does and you're not having problems with anything other than SSB. Everything you're saying points to a non-linear amplifier which should not be used on SSB. Sorry to be the bearer of bad news.