I would say it could be several things. First, I own a CSU carb, but I know C&S makes a good blow thru, so check with them first. The first things I would look at would be:
- the accelerator pump needs to be addressed, ac pump cam change
- if the carb has a PV it may need to be adjusted to when the boost is pushing it open, in your description I would say too soon?
- if youre using a MSD BTM and it stumbles when its coming up on boost, disconnect the boost timing retard and try running without it
- also, pull the hat off and run the car with no turbo boost, just start it on the motor and see what happens when you give it some throttle, that will eliminate the BTM and PV items for now