But try a 'large' capacitor in series with the solenoid. There will be a value that should work.
You would need some way to discharge the capacitor quickly when the trigger is released. If the trigger is a SPDT type, that might be possible. Otherwise I don't think that's the right approach. Also, using a capacitor won't give cleanly defined ON and OFF transitions to the solenoid; I'm not sure whether that would be a problem or not.
Try a switch operated by the solenoid. When the solenoid has moved its armature to the full extent, it operates a switch to break the current.
That means the solenoid would deactivate as soon as it reaches its fully open limit. I think the solenoid needs to be opened for a settable length of time.
Here's my suggestion. I didn't really want to use a 555 but it's a pretty good fit for the application. I've included Digi-Key catalogue numbers for all the parts.
When the trigger is pulled, CT is initially discharged, so pins 2 and 6 of the 555 are at 0V. Therefore the 555 drives its output (pin 3) high. This forward-biases power MOSFET Q1 and energises the solenoid.
CT then begins charging through RT. The 555's upper voltage threshold at 2/3 of the supply voltage will be reached after a time that can be calculated by multiplying the value of CT (in farads) by the value of RT (in ohms) then multiplying by 1.1, with the result in seconds. This formula is given on the schematic as well.
With the values given, the time period will be 0.11 seconds. Vary CT to vary this period by changing the value of CT, to suit the solenoid and the desired activation time. I've specified a capacitor with 5% tolerance. You can also vary RT to adjust the solenoid activation time, but keep the value between 68k and 150k.
When the rising threshold of the 555 is reached, its output goes low and the MOSFET turns OFF, deactivating the solenoid.
When the current flow into the solenoid is interrupted, the solenoid generates a voltage due to "inductive kickback", which brings Q1's drain positive. This voltage is clamped at about 48V by D1 and D2. Traditionally, a single diode (D2) is used to clamp inductive kickback, but that approach slows down the deactivation of the solenoid noticeably. Adding a zener (D3) speeds up the deactivation while still protecting Q1 against overvoltage.
When the trigger is released and power is removed from the circuit, R1 ensures that the circuit's power rail falls to zero quickly. This is needed to ensure that CT is rapidly discharged, through D1. Without these components, CT would remain partly charged between consecutive trigger activations and this would shorten the solenoid activation time.
C1 is required to decouple the 555's power supply and ensure reliable operation. C2 decouples the sense voltage of the 555 and is also recommended for reliable operation.
I've included Digi-Key part numbers for all components except the battery, the trigger button, and the solenoid. I've included three options for Q1. The FQU20L06LTU is a fairly cheap device with good performance in a small through-hole IPak package. The HUFA76429D3 has a lower ON-resistance (i.e. better performance) in an IPak package for about 50% higher price. The IRF510 has poorer performance and is in a larger TO-220 package, but it's more widely available.
The circuit can be constructed on a small piece of stripboad. The parts of the circuit shown with thick lines carry heavy currents and should be constructed with thick wire to minimise resistance and impedance.