Mike(FLYFISH) since the issue with your bike not wanting to start with the the choke out is present with both cdis I suspect that the cdi is not the problem. You should check to make sure that your gas tank vent tube is not pinched, make sure that your petcock is not gummed up with junk, and check the hidden fuel filter in the T fitting between the carbs. The "chokes" on our bikes are not actually chokes but fuel enrichers. When you pull it out it pulls more fuel into the carb, if you have a clog or obstruction pulling the "choke" or enricher out may be creating a vaccum effect and actually cutting fuel off to the carb instead of pulling more into it.

2006 865 Neon Blue/Jet Black Speedmaster,790cams, TTP Stage 2 Firestarter, Epco Exhaust with 10.25" Samson Vrod baffles, UNI pods with 40 pilots, 150 mains,Thruxton needles, 3 turns out.