Moving in together is a huge step in your relationship and is like to put pressure on it, at least in the short term. You need to be able to discuss your families needs with your partner openly whilst listening to what he needs also. If you are unable to do this, moving into together at this stage is very possibly not right for you .
Make list of what YOU want to cover in the session. Mediation is usually about custody arrangements as opposed to organising finances. Ideally you want to create a parenting plan so you both know where you stand and what the plans are .. this is also really helpful for the kids so they get into a routine.
I had this issue with my ex getting both our sons phones at a time I felt was too early. We managed to reach a type of middle ground where we put a few rules around the phones that applied to both homes although in reality I don’t think he kept to them and now they are older and I just work within the rules I make for my home when they are with me. Phones really do make life difficult when they are young. Hope you manage to find a peaceful way forwards.