This is almost exactly the problem I just fixed on my 83 GS550E. Welcome to the goofy technical problem club.
My first thought was a grounding problem. I checked with a couple local mechanics I know and they both agreed that it was distinctly possible that once the engine gets hot that a grounding point may lose contact.
What threw me off was that my starter solenoid burned out, with the starter side wire actually glowing red (wanna talk about a sight that will freak you out). I replaced the solenoid, but the original problem still was not fixed.
I finally gave up and picked up a used starter from ebay for a pretty good price and the whole thing works like a champ now.
As long as you're hearing that solenoid clicking, take it totally out of consideration for troubleshooting.
The starter is challenging, but not overly hard to get out. Here is my method I used a couple weeks ago:
-Remove battery and battery box (this requires removal of fuse box and two electronic parts on the other side of the box).
-Remove screw on top of airbox and bolts holding down the tank, then remove the hose connected to the top of the box and loosen up clamps on the carb boots.
-Slide air box back as far as possible
-Remove boots that sit between airbox and carbs.
-The choke cables will cause some difficulty, but I simply removed the cable from the choke slider on the left handle to give slack.
-Remove fuel lines from petcock
-Remove carbs to the right side of the motor and rest them on the right side of the block (unless you have something better to support it handy)
-Remove cam chain tensioner.
-There was a lot of oil and debris sitting on top of the block, so I used this opportunity to clean up as much as possible.
-Remove starter cover
-Remove two bolts holding starter to the block and remove the wire running from to the starter
-Here's where it gets tricky. The oil lines will get in the way, so as you are backing out the starter, you will have to turn it slightly so you can angle it around them.
I found when I was putting everything together, the boots were the most difficult thing to get seated properly. After much frustration, I found that installing them on to the airbox first then use the airbox to push the boots onto the carbs worked best. You will probably be wishing you could grow an extra arm or two to get the boots on.
Best of luck!