Just as spider hinted on: you need a timer to control the dwell. And if you are going to put the work into that circuit, you would be better off using one that already exists to do the hardware and software work for you.
UNLESS you just want to try it,.... Then "it's all good"
But spider is correct. Most questions asked about dwell with solenoids rarely look into the timing issue with the total voltage dump versus the timing that occurss with the reset of the valve and sear.
If you: used a micro switch, cap, battery, voltage reg., and a relay to dump the charge,.... Your dwell would be based on the speed of your fingers. And as you have pointed out,.... Your marker would chuff all the time.
Would it work? Yes. Would it work the way you want it to? No.
What I would do in your case is to get some bread boards from JAMCO or whomever and tests your circuit outside the frame. Bread boarding is a very cost effective way to use PCB parts that you can move and place with your fingers and test with little to no cost if you are "learning".
Packages for components such as SIP AND DIP work great to learn with.
So lay your gun on the table, trouble shoot your circuit outside the gun,... And test it as you modify the circuit.
You should be able to power it on, move things around and try it out. No soldering......
Jai "P8ntbal4me" Menard