360 motor loses ignition when warm

It's an electric conduction problem of one type or another. Which means in your case with electronic ignition it could be the coil, ECU, or the ballast. Problems like that can be hard to find because usually the part causing the issue will test just fine when cold. But when the part heats up its getting out of spec, either conducting too much or too little.

I had a problem with MANY brand new condensers on a dual point mallory. As soon as the car warmed up the condenser would stop working and the points would fry. I practically went through a case of them before I found a one that would work. Quality control on stuff like that sucks nowadays. And of course, they test fine cold, so they're hard to catch unless you're actually heating them up.

With the electronic ignition it could be the ballast, coil or ECU. Those would be my prime suspects. Ballasts are cheap and fail pretty frequently, easy to replace that. Coils are a little more, but even just a quick substitution of a known, used coil will tell you what you need to know. Modern made orange boxes are junk. I've had issues with several brand new orange boxes. It would be my prime suspect in your case, but they're more expensive, so it may not be what you want to replace first.