I went through HECK with my K 1100LT
I set the testers, probes, meters and all my own theories aside and tried something I hadn't yet.
I had the fuel pump and fuel sending unit outside of the tank sitting on an old pair of jeans with the fuel level arm sitting at about a half tank. I then powered up the bike as though I were going to start it, no fuel pump and no fuel gauge. I'm on the Gremlins tail. I grabbed the four pin connector and twisted it all around and watched the fuel gauge jump from empty to full but still no fuel pump. Note worthy it is that I had jumped the side stand switch and the gauge flipped from full to empty.
I replaced the four pin connector for the sending unit with male and female GM style water proof connectors and the bike has ran fine without even a hiccup since that simply repair.
Have you ever sat behind an automobile towing a boat and trailer and wondered why the tail light on one side dims or goes out when the brake lights light up? Its a bad ground, not a complete ground disconnect but one bad enough to let low amperage work but when you ask for additional amp flow like a brake light or a fuel pump to use the ground it just isn't good enough to do the job.
Replace the four pin connector with suitable replacements(not a trailer connector)and you very well save yourself some cash and lost sleep.
You could very well have two seperate problems
I would replace the four pin connector and eliminate the the possibility of it being your problem. If it works after that I would and your spark returns and fuel pump works then I guess that connector could affect the spark also.
The manipulating of that connector seems to be a common denominator in the no spark & No fuel issue.
I would also be prepared to investigate the HES as Paul suggests, its not his first rodeo ya know. That connector could very well carry the ground circuit for the HES.