Yep, 100 percent acrylic paint will stick just fine to the plastic mouldings surrounding the windows...I never use primer and I have been painting those doors for 20 years. But go ahead and use a bonding primer if you are worried...priming is never a bad decision. Don't use TSP because it is overkill and it has to be rinsed. Paint thinner will work just fine, followed by a quick sand with a fine/medium grit sanding block. You do not want to sand so hard that you remove primer and expose bare metal...just a quick, light sanding.

I have one more factory primed door to paint and I will sand and add extra primer on that sucker before rolling color onto it. It was a PITA to re-do the front door. The paint peeled off the entire door. Paint man said he has more people having to repair that mess as the primer on these doors is crap. I believe it. Anyway, things are good with the door now.