Either the choke is adjusted wrong or the choke pull off has a hole in the diaphram and is not opening the choke a little bit just after starting.
Is it rich or lean when you are having the problem?
You can check this by starting the engine and opening the choke or closing it part way, the engine will "tell" you what it likes.
To check the choke pull off, look on the right hand side of the carb and there is a vaccum can attached by a link to the choke blade. Pull the vaccum hose off the can and compress the arm coming out of the can.
Put your finger over the vaccum hose port connection on the can, and release the arm. The arm should stay in until you release your finger from the vaccum port.
It may come out just a little when you first release the arm, but not all the way until you release the vaccum port.
If the arm returns with the vaccum port covered there is a hole in the diaphram.
This will prevent the can from opening the choke blade about 3/8" of an inch just after the engine is first started. The engine will run rich until the choke warms up enough to open the choke blade.
If the engine is dying in the first 20 to 30 sec only, and the choke pull off is OK, then you can bend the link to the choke pull off to make it open the choke blade a little less or more.
I think making it a little longer will help the most for your condition.
The danger of richening the choke to make the inital idle better is perhaps making the engine run a little too rich after the first 30 seconds. You'll have to play with it each morning you start it up.
If you still have the instruction sheet for the carb kit read up on it there.
Check my web page for suspension info:
67 RS 327
69 Camaro Vintage Racer
65 Lola T-70 Can Am Vintage Racer