No easy answer Rachel. Yes...absolutely do NOT bring her up. That is a cheeseless tunnel as they like to say around here. It will get you nowhere and only increase the tension and resentment between you and your H. And DO NOT check his phone...not under any circumstances. Another cheeseless tunnel that will only hurt you in the long run. Focus on your daughter and do not waste time watching him and trying to figure out what he is doing.
Also...I don't know the exact details of your arrangement with your H in terms of him being at the house but is there any reason he can't just come over and take her out? Does he have to stay with you or be in the house? I would also steer clear of family outings for now too if I were you. I know it feels like it is better for your D but it really isn't. It sends confusing messages and it is also really, really hard for the LBS to maintain their "everything is okay with me" stance when it is such early days and there is an AP involved. IMO, it is needless self-torture.
I remember early on in my sitch, my XH had our kids out on our (now his) boat. I was at home and had plans to do some yard work and go to my sister's for dinner. He texted me to see if I wanted to join them on the boat to go check crab traps. If I had fully understood DBing at the time, I would have politely declined and went ahead with my plans. But...I really wanted to spend some time on the boat and at that stage (first few weeks) I still had hope that we could reconcile (he was not admitting to OW and was still calling it a "break") so I said okay. The boat ride was nice, felt like old times, but then afterward he dropped me off at my car and then took the kids to the pumpkin patch as he had originally planned. In my head, I had pictured us going out on the boat and then him inviting me to continue on with him and the kids. When that didn't happen, I was devastated and I ended up sending him a nasty text that I later on regretted. He apologized and seemed quite shocked that I was upset...said he had felt GUILTY about being on the boat without me and was "reaching out" (translation...to alleviate his guilt, not because he wanted me there). Anyway...I learned a big lesson that day. The first was to stick with my plans and the second one was that I needed to let go of any expectations I had of him and his behaviour towards me. I also learned that I felt much better by myself than I did when we were spending time as a "family" because the whole time I felt like it was a lie and I would either feel sad or mad or some combination of both and it would end up being a big setback.
Only you know your situation and your own level of tolerance so you should do what you think is right. Just take my advice for what it is worth...from someone who has walked a mile in your shoes and made all kinds of mistakes. Also someone who, two years later, is divorced and loving her life. I NEVER for a second thought I would get there when I was at the stage you are at. Hopefully just knowing that you can survive and even thrive after something like this is of some comfort to you. (((HUGS)))
Me 51 H 46 B/G Twins 11 SD19 Legal SA - January 2019 Divorce filed - June 2019 Divorce final - November 2019
Together 14 years Married 12 years BD1 - May 2014 BD2 - September 14, 2018