It takes a decent amount of pressure to propel BB's. Are we talking old school shoot-your-eye-out metal ones, or 6mm plastic Airsoft pellets?
An Airsoft pellet can get decent speed out of lower pressures. Small metal BB's, not so much. Basically remember the force from 28 PSI is proportional to the area- a projectile in a 1 sq in cross-section area barrel will see 28 lbs of force, a projectile in a 0.1 sq in barrel gets 2.8 lbs. The smaller metal BBs see less force and are much denser too.
There are already electric airsoft guns. The motor pulls back an air cylinder on a spring, then lets it go and it compresses air rapidly. No air storage necessary.
Other notes:
A compressor may take a significant amount of current.
A problem you may run into is that you're dumping all the air you've compressed which is terribly inefficient. Like, ideally you'd want to only dump air close to 28 psi and then just stop the flow when the projectile leaves the barrel and the pressure's still at 25 psi. If you dump it all, then it doesn't make it go any faster but it takes forever to rebuild the pressure from 0 psi.