This is an interesting question. Before finishing the entire description I would have opted for your technique is not up to par and you are getting fatigued. However, you point out that this does not bother you at home and that it starts right away on stage (after the first pinch).
If this is solely a performance anxiety issue it's probably psychological. You are nervous and that causes muscles to tense. Once they tense it's a vicious cycle!
After one mishap the mind might go to one of two places in my experience. (1) I need to repeat that last bit and get it right, or (2) oh no it's falling apart I hope the next one isn't so bad.
Neither of these is a useful mind frame for performance. If this is a fair evaluation of what's going on then I can relate as I have had varying degrees of stage fright in my life. One thing to consider is that if you are playing solo (no back up) then you really are naked (and possibly afraid [couldn't resist]). What helps me is working with bands either as the front or as back up. If you can get on stage and back someone else up for a song or two that may give you a chance to acclimate to the climate and relax a bit while someone else has the heat on them. Sorry if that sounds selfish but it's true. Of course you need to do a decent job as a back up or all eyes are on you, but in my experience this is a low key task. It puts you in front of the audience for a while in a non-threatening way and hopefully in time your nerves will calm down.
It has been my experience that my stage fright peaks right before the first song but then completely vanishes. This is probably due to years of live playing. The calm during the storm is critical for job security. I do it for a living and if I had the problem you are describing that would not be possible.
Sight readers have the problem of wanting to stop and go back when they mess up and that is not possible during a gig. One thing that helps with this is to do regular sight reading exercises. There are good books by William Levitt on this. The goal is not to read a song to learn it but to set the metronome at a decent speed and NEVER STOP, even if you F@$k up. Always stay on beat and either stop and jump in on time or recover as gracefully as possible. I am not suggesting that you have a sight reading problem but this method can help you relax during performance. The same tension builds (or used to build) when I played in large orchestras and missed a cue or felt like I might fall out of sync with the group. That mishap causes extreme tension and loss of confidence and eventually an inability to move forward. You could try the same approach when practicing at home by simply pretending you are on stage and either playing with a metronome or a recording of the song(s) you want to perform. If you mess up, let it go, DON'T STOP just pick up ASAP. The flow of time will not stop for you, or anyone, and that is one thing live performance teaches you. Keep track of how well or poorly you do with the pick up and try, try, again. You will find that even though mistakes come you are better able to handle them gracefully.
Now on to technique. Based on what you have said I'm going to guess that your technique is fine but it is worth mentioning that if you have any bad habits, or poorly developed techniques, or even if you don't know the songs you are playing perfectly, these will all come out amplified by 100x's when you get nervous. If your hand is cramping to fatigue you are definitely squeezing too hard but my question to you is, do you practice long enough at home to know when you will fatigue? Believe it or not many guitarists practice bits and pieces of songs, and get really good, but don't rehearse the entire pieces. On performance day they might fall apart 2 minuets into a 5 minute song because they haven't tested themselves. Be honest with yourself. If you have noodled with a tune and really think you know it make sure you are playing several times in a row without fatigue before attempting it on stage. If you cannot get through it w/o a hand cramp or other issue at home it will not happen on stage. And again, nerves will make it worse. What I'm saying is make sure your body is really ready for it. That will help reduce the amount of tension generated by stage fright. It's all about reducing risk. If you know the song inside out and don't fatigue you might be able to let your body take over. If you cannot get through the tune several times at home then you are probably not ready. If you start getting hand pain you may need to evaluate your technique and improve it.
Last but not least on the stage fright front usually playing more makes it go away. But if you get the same type of anxiety talking to people in tense situations, like returning something at a store, giving a presentation at work, you might want to try regular meditation or TM. The long term effects are that the tendency to get anxiety simply goes away in all situations.