If it is a vacuum secondary it will have a diaphragm on the rear of the passenger side of the carb. It will have a removeable top and springs. Stiffer springs will slow the opening. If it's not new, but "Modified" by some expert, go back to the original spring. That will normally solve the problem.

        At this point I'd check float levels and make sure the correct metering block and or jets are installed. Carefully inspect the gaskets and make sure there's no air leaks. Make double sure all the unused vacuum ports are closed/capped. You might try blocking off the secondaries with a rag or tape to sure the issue is in the secondaries. I'd check the fuel passages to the secondaries carefully now also. The thing that bothers me about your description is that by the time the secondaries start to open any pump shot you have is long since history. This sounds to me like a primary side lean issue. Low float level, leak in the metering block gasket, Vacuum leak, plugged accel pump passage or shooter.  

Don't forget the ignition timing also.  The Demon carbs are partial to more ignition advance at idle due to the leaner fuel circuitry and this is a player when accelerating.  Advancing the idle ignition timing may require shortening up the timing lead in the distributor to keep from having too much total timing at rpm.