Is there a way to fix a small plumbing leak from inside the pipe?
I know a lot of industrial non-permanent fixes, and methods of investigation but there is to my knowledge no surefire way to do this permanently and solidly without opening it up, soldering, replacing or clamping.

An example of a temporary fix - you can maize it. That is what the mechanics at work do when there is a leak. (it is not easy, safe or practical to shut down a 50MW industrial furnace for smaller water leaks, but stop once in a while and fix them all) 

I was astonished, I thought this was some high tech stuff, but it is quite literal. They add cornflour to the circuit. (a degenerate brandname of corn flour around my parts is "Maizena" - hence they call it "maizing"). If the leak is rather big-ish (like a faucet at full opening) they add corn flour and sawdust. If it is any bigger than that, people get nervous and the yellow-helmets start rushing about.

But, to reiterate - no, a leak should be mechanically investigated and fixed. You cannot know if the source is a pinhole leak, a bad fitting or even where the leak or fitting is that is leaking. Dripping only signifies where the water is leaving the surface of the pipe and hitting another surface. It could be leaking somewhere else. It could be leaking all the time, only dripping when you run the hot (running water has less pressure. Maybe it is spraying constantly). Water where it shouldn't be is a serious problem - it can cause a host of unpleasant second order issues (mold, rot, rust, cracks, bugs, electrical shorts... etc). There are plenty of temporary solutions, but few proper ones.