I have one of the connectors near the battery have the female connector side loose and causes an intermittent power loss.
What's the preferred method to fix this? Are new lugs/inserts available?

Just wire around the connector.

How many wires are with this connector? I know of EDIS connector,ECU connector. Some cars have this 2 wire connector (I think one red wire one blue) Last would be fuse block.

7. There are two the same size right next to the batteries and relays. It goes into the fuse block and then to the rear. I thought the relays we're going bad, but it turned out the green wire on the connector separated from the lug that inserts into the block. I got it out, soldered it together, reinstalled, but it didn't click into place and it failed again on the track. Wiggling the bundle re energized the circuits.

Rather than pull it apart, I wondered it it was legal to replace it with a mil spec connector...the gcr says routing and attachment as per rsrii no mods allowed but is silent on repairs or parts. I supposed a different connector would be a mod, yes?

This not a connector that you ever take apart. The new kits don't even have it. Just wire around it and forget it. Nobody will care.

So you are saying green wire to relay. You can cut it out, butt connect in a different one. What I'm saying is cut your wire from block. cut off the block from another harness and splice it onto your wires. Before you connect to wires put on some heat srink tube long enough to cover your connectors. All will be good. You can also replace the harness. Remember the harness is cheap due to Gen3 change. They are laying everywhere. I have 3 myself.