I would not worry that the typical polymerization inhibitors added to acrylic esters will interfere with the aza-Michael reaction. I guess you will add the amine in excess, and the relatively small amount of the inhibitors, if they react at all (which I doubt), will do no harm. Depending on your reaction conditions, you may even want inhibition of the radical polymerization, for example at elevated temperatures.
I also assume you will have effective workup procedures to isolate your products in pure form, so you may want to pay attention to remove the inhibitors then so that they will not disturb whatever you plan later on. After the double bond is lost due to the reaction, the inhibitors are not needed any more.
Something else to keep in mind: Your reaction products are $\beta$-aminoesters which are much more prone to hydrolysis or alcoholysis than their counterparts without the amino group.
I have used various acrylic esters successfully in the past in Michael-type addition reactions and never bothered with removing inhibitors first.