Does your 1911A have the battery option? If so, the batteries need to be installed, even if you are running it off of AC. In fact, check the -12V out of the inverter.
I actually got two of these units in an eBay purchase. One was completely inoperative and the fault was isolated to one of the custom controller chips (U6, or U7, I can't remember which). It was easy to isolate because I could switch the chips between units and watch the fault follow. I know that's not particularly helpful in your case, but it is a warning that the custom chips can die - and that's pretty much the end of the story.
I would not spend any time, effort, or especially money, trying to fix the 1911A. It is, in my opinion, not a very good counter. I would look around for something else. There are cheap new Chinese models and lots of used (and perhaps broken) HP models for well under $100. Fluke was not known for their counters, although the later ones made with Phillips aren't too bad.