I've never experienced "cutting out" in connection with anything hot at the fuse box. I'm inclined to say there's no connection, but others may have a different view.
First, on cutting out. Cutting out in my TR is almost always one of the following: 1) Improper carburetor mixture, and materially so. Minor instances of too rich or lean will not do that; 2) bad points or distributor cap (I can't comment on the condensor); or 3) bad plugs, possibly fouled, or bad plug wires--even to the point of arcing.
I've learned never to overlook the plugs, plug wires and distributor cap in that regard. I once noticed that the plug wire to one of the pistons was "arcing" to the fuel line causing missing and stalling at higher RPMs. In another instance I had a badly fouled plug from running too rich.
I'd pull, inspect and clean the plugs, check how they snap back on, verify nothing is arcing to nearby steel, and check the distributor cap and the ignition wire leading to the coil. That wire very frequently is frayed, deceptively so, and goes unnoticed. It is critical to getting current to the coil and distributor.
If your carbs are adjusted right, and your plugs and wires are correct and intact, I think the "cutting out" will be solved.
A last issue might be blockage at your fuel tank or in the fuel line. Again, that can be more frequent than would be expected. I once had a piece of wood floating in the tank that intermittently would block the outlet, and you might imagine how long that took to discover.
Back to the hot wires. Electrical issues can require the greatest of expertise, and I won't say I'm anywhere near qualified, but here is some advice from experience.
When the ammeter wire to the fuse box is hot, you have to stop and find the cause because it's serious, just as you've suspected. If the ammeter blows, everything will seem to be dead.
You should disconnect the battery and carefully check all wires where they meet at the ignition switch and be sure you don't have any frayed wires or any short circuits due to wires touching others, or interfering with any nearby steel, such as on the dash panel. I'd be suspicious of any accessories added, and how they are connected.
Check for bad wire splices and get rid of any crimp connectors or plastic splice fittings and replace all of those with properly soldered connections with heat shrink wrap over those. At least one good mechanic told me that British car wiring is too sensitive for Auto Zone and Checker dummy snap fittings, crimps and all the rest.
Check the wire in and out from the ammeter gauge and definitely check for shorts at the gauge. Make sure the gauge isn't fastened such that either of the wires connected to the back is shorting out with the dash panel or anything else. I had exactly that problem at one time.
I believe one of the ammeter wires goes directly to the starter solenoid. Check that carefully as well.
Finally, I'd check your fuse box, itself, to be sure none of the other wires is incorrectly located and none is shorted out. You'll need a wiring diagram and sometimes they are hard to read.
Don't laugh, but if none of that works, I'd start wondering about the voltage regulator because, other than assuring the wires are all connected properly, that one is a mystery to me and I've never heard of one being bad--though I'm sure it happens.
Good luck.
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