Some time back, I hooked a 12v 400w 120v automotive style consumer grade square wave inverter to a big car battery, put a bridge rectifier on the outputs, and:

With bridge rectifier on output, I measured 140v dc. When I tried it with a 2200uF smoothing capacitor, I tripped the safety overload
circuit. Is there an easy way to stop this from happening? Maybe inrush protection circuit of some kind?

If I try to use this style of inverter with a transformer on the output, what kind of efficiency will I see? Will the transformer overheat?

 

Some time back, I hooked a 12v 400w 120v automotive style consumer grade square wave inverter to a big car battery, put a bridge rectifier on the outputs, and:

With bridge rectifier on output, I measured 140v dc. When I tried it with a 2200uF smoothing capacitor, I tripped the safety overload
circuit. Is there an easy way to stop this from happening? Maybe inrush protection circuit of some kind?

If I try to use this style of inverter with a transformer on the output, what kind of efficiency will I see? Will the transformer overheat?

Upon reflection, this is a complicated subject, and my two-part question doesn't lend itself to a straightforward response.
After doing some more reading, I'm thinking about going a different direction.

1st, the inrush is tripping the protection circuit in the inverter, and so I may try an inducter in series with the output load (rectifier-bridge+capacitor), but this maybe insufficient, and require an active circuit to limit the surge at start-up.

2nd, the efficiency will be 80% or less, as heating will be an issue in the primary, even before loading the transformer, because the low frequency square wave on a iron core transformer will likely saturate the core with each half of the cycle. I may try both and inductor and large capacitor in series before the transformer, to ease the voltage spikes/second order harmonics in the primary. It's appearing my original idea is neither simple or easy.