I use a USB powered 555 relay trigger board in conjunction with an Ne555 signal generator. It serves two purposes:
First, it provides a reliable way to trigger the ne555 signal generator. You don't usually get enough power from a light gun's trigger micro-switch alone. You can also use a larger power supply for the signal generator. I use a 9v 1a psu.
Second, it gives you a 3rd pot to control how long the signal generator stays energized. With some careful adjustment, I am able to make it trigger once when I tap the trigger and full auto if I hold it down. This works well for 99% of games, there is no switch needed when you change weapons in Time Crisis 3, for example. Like this:
You don't necessarily need to use a relay trigger board though. You just need a reliable way to trigger the signal generator. Other people use a mosfet or a Pac drive controlled with mame hooker. In your case, you can use the output from the Aimtrak recoil board but with a smaller 5-9v psu.
I also use a DC DC solid state relay between the signal generator and the solenoid. I recommend that you do the same. I've had great results from ones like this:
https://www.ebay.com/itm/SSR-100-DD-Solid-State-Module-Solid-state-Relay-DC-DC-100A-3-32V-DC-5-60V-DC/312100885769?epid=2245902400&hash=item48aaa99509:g:c1MAAOSwwBNawaBSI discovered recently that those little signal generators are only good for 30w or less. I managed to melt 3 of them trying to power a 12v solenoid directly. The last time, the ne555 chip literally snapped in two with an unnerving bang. Do not power the solenoid directly, even for short testing....
If you have problems with those little red signal generators, I suggest the larger blue ones I posted links to. The two red ones I started with didn't last long. That doesn't mean yours won't but if you have issues, don't assume it was something you did wrong.
BTW, the Aimtrak solenoid that comes with their kit is not rated (or suitable for) full auto. You need one rated for continuous duty. The Aimtrak one is only for momentary use. Mine lasted one game before melting all the wires and breaking. You can use it for short test runs but keep the voltage less than 24v and keep the duty cycle low.