Although you have 12.6 volt at connector without load , it could drop out once unit is connected.
Worth checking / replacing fuse in fusebox for corrosion
as this could cause problem
you describe.
Good luck!!
Took the wife fishing, never got past Branksome, very, very lumpy in the stronger than forecast South Easterlies !!!
Retreated to harbour for some relaxing tiddler snatching for us....no bream caught.