Yes, strange and aggravating things may happen, but it'll still work. One BIG problem that's gonna show up is starting the arc. CC power supplies have a low short circuit voltage. That means your gonna put the gun in position, pull the trigger, and the wire will advance towards the metal, when it gets REAL close, the level current output from the source might (will) blow away all the wire back up to the contact tip.

The fix is to have a weld controller between the power source and the wire feeder head. Look to Hobart or Miller for these. All they do is when you pull the trigger, the arc starts, and before any burn back can happen, they accelerate the wire speed,( this is like split second stuff) and maintain the correct wire speed regardless of any voltage drop-off from the source. They also have a relay to turn the output voltage from the source off to keep the gun cold when you let go of the trigger. I wouldn't use a contactor to switch the supply voltage to the power source on and off unless you don't like it. The inrush current would eventually kill the rectifier. Beside's, the thing will sound like a train wreck every time you pull the trigger.

This is why I prefer 99 different machines, each one has a specific job it was designed for, makes the job easier.