Boondocking - power transfer switch (DC to AC ) quit working

No you are not going to charge your batteries until you get AC power inside the rig. It concerns me that you posted (DC to AC) as a transfer switch does not do that... it switches the AC between shore and generator.
Sounds like the coil that holds that side of the contacts closed is not getting power. If you cannot find the problem (are you sure you have power from the generator), I'd figure out how to manually hold those contacts closed or run jumper wires around them, or bypass the transfer switch completely... for now.