Hmm, balance is definitely something it sounds like you guys need.
She is very young to be the mono(?) in her first polyship. Lying about your age (heck, about any major personal detail) is a major red flag for me, but obviously your partner has made his peace with being lied to.
Are both of them unwilling to compromise on time? Say, you get two date nights per week and she gets two. The constant texting can be extremely annoying, and I think you are fully within your rights to ask him not to text her for the few hours a week you have designated as 'just you two'-time, but what's with the 'not in my house'-thing? Do you live separately and he comes to hang out at yours and keeps on texting her? That is just bad manners. If he lives there too, isn't it his house too? How long have they been involved? The worst of NRE can take a few months to cool off. And if you can't wait for it to slow down on itself, you can always try texting him yourself
. If you do it from the next room it should alert him to you feeling a bit neglected.
I would suggest agreeing on certain week nights beforehand, or if you have too versatile lives for that, to agree on a Sun afternoon who is going to get which night next week. And to not change plans unless there is a major life crisis (i.e. someone is in the hospital). Although some people dislike fixed schedules per se, you can't really afford that if you are trying to handle multiple romantic relationships at a time.
What were bf's motivations as to continue his relationship with her even after the age hoax was out in the open? And for yours to back away? I fear that you would be happier if you started to pursue relationships with both men and women on your own rather than trying to find the perfect match for the relationship you have with him.