I have never had SPDs unclip while riding fixed, though my old 505's did unclip a couple of times on my geared bike before I learned to set the tension correctly and not turn my feet if i didn't want to unclip. Doing these two things is key to success with any clipless system and you probably know how to do both already. There may not be any need to buy a different set of pedals plus a new set of shoes for your fg.

As a p.s., apparently the silver Shimano cleats are no good but the black ones like I have seem to be no problem.