I can understand how OP's partner is feeling, any man in his shoes likely to feel that way but he's being emotional and vindictive in how he's dealing with it and has had issues about dd1 since dd2 came along.
The problem is that whilst he might resent dd1 (like it's really her choice..) if she goes to private school and they don't end up having a third, equally there's chance OP will resent her partner if she has a third and can't send any of them to school she prefers or he ends up alienating dd1 even more after his dc2 comes along, regardless of which school she ends up at.
OP, whatever you do, the issues with your partner will not go away, he's had them since dd2 came along so even if you don't send dd1 to private school, he'll continue his resentment of the so called imbalance in the time you give her and money her dad spends on her.
Almost all first born children end up getting more attention from their mums in particular, it's kind of inevitable, everything they do is new for everyone and needs more focus, they're the guinea pig, your partner doesn't have a benchmark but I would say I have the same issue with my two, my eldest always ends up getting more of my time. This situation must be incredibly stressful for you if you can't be a natural parent for fear of causing resentment in your partner.
Since he's taken the extreme step of stropping off to his mums, can you suggest having a session with a counsellor where you can both speak openly about the issues, I think your relationship is on on a knife edge and sounds like it's going to need some major compromise on all sides to survive. A neutral third party might help you to thrash it all out?
In the current circumstances fair to say the only person preventing you both from thinking about a third is him, he's behaving in a way which makes it impossible for you to contemplate a third with him and this has nothing whatsoever to do with choices you plan to make for dd1 but his inability to acknowledge what a fucking difficult situation it is for all of you and emotional blackmail is not what's needed.
Your dd1 is in a lose lose situation right now, totally crap for her and that will impact on dd2.
If you and your partner want to sustain your family unit, you'll have to sit down together to talk about yourselves as a unit and what works best for all of you in order to survive. Personally, I think trying to create two family situations under one roof, will be polarising give the background.
Sorry, long post, hope you can get some counselling, this is going to be a really difficult one to work out without some neutral, calm, help as you're having to deal with some entrenched feelings on both sides.