The rubber disk is just a push fit in the recess in the end of the plunger so should not easily drop out, especially on a new carb – this makes me wonder about the ‘quality’ of your new carb and whether this is really the problem!
Dry the plunger and disk and use a tiny drop of superglue perhaps to hold it in place.
When the bike is running (if you get that far!) and the choke is in the off position, the plunger with its rubber disk is spring loaded up against the choke fuel feed opening so it can’t just drop out.
You say the spark plug is dry, but if the choke plunger disk is displaced or missing then the carb would flood constantly and there would be fuel everywhere so that is probably not your main problem.
There are reports about the variable quality of aftermarket copies, some report no problems and some report that they just put them in the bin.
I bought a new MZA carb (with the BVF logo cast in the body) which was more expensive than some of the ‘chinese copies’ and have not had any problems (so far) but bear in mind that back in the day, even when new, the original carbs were fairly crude and wear quickly.
Did your bike work ok before you fitted the new carb?
Check fuel is getting from the tank to the carb?
Check fuel actually getting into the float bowl – flip the clip off?
Try a few mls of fuel down the plug hole and see if you get at least a splutter.
Let me know!