Engine dies when warm

That '70-'72 choke thermostat's pushrod was meant to hook to a Holley 1920 or Carter BBS (final bend at the carb end of the choke pushrod is to make it point forward). The Holley 1945 carb on your engine requires a choke pushrod with its final/uppermost bend making it point rearward. Bends below that are configured to clear obstacles and allow the pushrod to move freely without binding. Someone has rigged up your setup to have a choke pushrod that would connect to the 1945 (top bend points rod rearward)—probably by robbing the pushrod from a '74-up choke thermostat. But that's a different kind of choke thermostat, and the pushrod length and angles are different. That's probably one of the reasons why you're now having difficulty. It's also probably why the cap is missing, which adapts the small choke pushrod to the large hole in the choke thermostat housing (required to allow the rod to move in accord with the arc of the carb's choke lever without binding down at the thermostat). This cap is there to trap heat so the choke won't re-close too soon and cost you gasoline money.

That '70-'72 setup, with the open-hole mount for the choke thermostat/cup/gasket, is prone to exhaust leakage and other nuisances.

In your shoes, I would fabricate a blockoff plate and gasket to go in place of the present choke cup, and perch a № 1234 electric choke kit atop the plate. End of all choke problems.