Here's my thoughts... and feel free to disagree with them, but this is from my own personal opinion on things and keep in mind some of this is hitting home for me in some ways, so I'm sure that has a lot to do with it (so grain of salt and all that)
Yes, he needs to do poly his way, and he's not going to act like you or do things the way you want. You're doing great to see this, and to see that part of these issues is you wanting to control things yet knowing you can't. I so get this!!
But... there are a couple of red flags here. He lied to her and is continuing the lie in order to get stuff. Be it sex or things, he's reaping the benefits of this by using her, and lying to do it. Even if you two were not together and he was a single guy out dating, this would be not kosher. It's not okay. And that is the most important thing here, I think. She is being used for sex and attention and has no idea because she's being sold a story.
As for the stuff, I tried to explain to my hubs that when a woman starts buying you a lot of crap it means she's in love with you. Hubs had an old GF he saw a couple of weekends in the beginning of the year while he was gone, and she started doing that after the first weekend they spent together. Sent all kinds of crap, something he mentioned liking and suddenly it's showing up at his place. I told him, she wants him back, it's not just casual like he thought (even though that's what they'd discussed). Hubs laughed me off on that one saying, no, she just "does that". Yeah. Next day he gets seven texts in a row about how she doesn't know how to do this, she has all these feelings resurfacing, etc. I was like-- no kidding, big surprise.
So he's basically told her you guys are separating, she's gotten the impression that he's being neglected, you spend all this time with your boyfriend, and then is surprised that she thinks she's "dating" him, takes him home to meet the family, and during all of this he thinks she's a plaything and might be done with her at any given moment.
For me, the issue here wouldn't be even poly/relationship related. It's the fact that he thinks it's fine to lie to a woman to get her into bed, act like they're dating or building something, all the while thinking he can drop her at any moment, no biggie, because he doesn't really care about her at all. That's something he needs to address, figure out, and fix. That's just NOT cool behavior anytime.
Maybe he just felt unbalanced with you being in a relationship first, maybe he felt left out, maybe he just wants some attention-- I get all of this, believe me, I do! I'm in the same boat with hubs. And that may explain the behavior, but it doesn't make it ok, and he needs to man up and realize that.
I think maybe you've been letting it go a bit more than you should out of guilt because he's had to do all of the work and you don't want to cut into any fun he might finally be having. You're second guessing your own feelings and intentions-- and I understand that, it's good to make sure you're coming from a good place and not just reacting. But I think you need to really look at the situation and sit down and talk to him about it.