He's not even 2, so I highly doubt he's pooping on purpose. Are you sure he has been reliably potty trained for many months now? Even kids who are trained can regress when there's something new or stressful going on (like a new baby or even a super-tired very pregnant mom, which you were when this started). But are you saying he was entirely potty trained at, say, 18 months?? Maybe it seems like he's doing it "on purpose" because he doesn't normally poop at that hour of the day, but extra activity and agitation can alter the schedule.
And he's in a toddler bed now, which he has figured out he can easily escape from. So lots of adjustments.
Go from the idea of "nap time" for him to "quiet time for all" and don't tell him he has to sleep. That's what day care centers and all-day preschools do - everyone lies down and if they don't sleep, they don't sleep. But many will drop off with the dim lights, darkened room, and lack of other stimulation/distraction. If you can, make this your special time with him with the baby happily in a bassinet or crib elsewhere. If the baby isn't yet on that kind of schedule, definitely make it "siesta" time for all. Get in bed together, or put him in his bed with you next to him in a rocker, and read something quiet and soothing. Read more and more slowly with a very calm voice, and see if his eyes start to close. Put on some very quiet music or even a white noise machine (we used an air filter that made a constant noise without the air circulation of a fan).
Locking a room from the outside if you are right there is one thing. Some people use a half door (Dutch door) or a gate. The problem is, if he's angry and frustrated, and he starts climbing on bookshelves or if he pushes the bed near the window, you can't control it or even see it. If you keep going back in, he won't calm down at all.
I'm sure you want to use his nap time to accomplish other things but it may be best to just accept that the dishes aren't getting done and neither is the wash, and just make it quiet time for you too. He needs it, but so do you. This will also get you into a good habit/routine for when the baby is older and gives up the 2nd nap for one good long nap. You'll be a pro by then!