You may be able to find a welding/cutting regulator and flowmeter combo, but the flow rate will likely be too high. You need more when using it as a shielding gas. The upside is, you may find one with a direct read scale for CO2 as well as mixed gasses.

If you do find just a reg, make sure the output ranges are within the operational limits of the solenoid and flowmeter (20-60psi is perfect). You don't really need more than 10psi on the output. You'll get more flow than you need...in fact, 10psi will max out your flow meter if you crank it up.

The combo I mentioned is around $70. Not bad considering it includes a solenoid that sells for around $30 online (and the other stuff you'll probably throw out).