I am not an expert of electric things but, this is what I would do regarding what you described.
I would go back and check all of the fuzes and breakers after the battery was hooked up again. Recheck the battery connections. If they are all good I would check the battery to make sure it is still good. It may not be holding a charge or cell(s) are no good. What is the age of the battery?
Than if it is fully charged and holding a charge, I would unplug shore power and see if anything works on 12v only. After these steps I would than think of a invertor problem. I do not know if there is a fuze at it, I would think it would be at the panel with the others. I did go to a workshop from keystone and they said that trailer systems need a battery hooked up to work even on shore power but, no mention was made if the battery had to be in working condition.
Please list year of trailer and it might help others help you.
2013 24RKSWE (27ft TT) Cougar 1/2 ton series SOLD 10-2021
2013 Ford F350 4x4 CC 6.7 engine, 8 ft bed, 3.55 rear end, lariat package
Retired from Oregon State Police in 2011 than worked another 9.5 years as a small town traffic cop:
As of 05-2020, I am all done with 39 years total police work. No more uniforms for me.