Oh boy. I don't know what I can say, FS, but we are here to for you to talk to. Vent to, if you don't want a 2-sided convo. Sorry this is so long.
The first thing that jumped at me was paragraph 10 about D13. He is projecting something - maybe his past? Guilt? Who knows. But he cannot read the future, and projecting something so bleak for your daughter is something you should immediately dismiss. Perhaps she would take it a bit harder than D10 (I don't know their personalities), but this in NO way ruins her life and condemns her to some future without prospects.
This reminds me of his disdain for piercings/tattoos and how he thinks one would eliminate D10 or D13 from any "serious" work or career success. I'm seeing a major hangup here for him and it irks me. If things aren't perfect than they are irrelevant? Does he feel that way? It ties in to his desire to not want to be the bad guy. And his desire for you to stay in the home, have no change, and remain in a comfy arrangement. "Picture perfect" only.
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When you say "solicitor" does that mean lawyer on this side of the pond? If so, that is wise. See if what H says about the house is accurate. Only make decisions based on real info. I think you have many questions to unearth on this topic. You're a very intelligent woman, so I'm sure you'll get a full picture of what a sale might look like with every scenario layed out.
Question: If it turns out that he is not part of the sale (because he is not entitled to the equity earned since he moved out, and he is considered a second/separate household etc. etc.) then why is he inquiring with agents to get the value? I realize it may be helping you, but if you are the one ultimately in charge of the sale (and living there) I think you should be in charge of the decisions made around selling. If it turns out the sale is on your shoulders you should be the sole one to decide what deal to take, and who you wish to work with etc etc.
Gosh, the child support thing. It didn't occur to me. But you are absolutely 100% correct to bring this up to him. I'm so sorry this is turning into such an intense and real discussion. I don't know if these points will put him at ease, but perhaps in a future conversation: 1) The solicitor's job is to assist the two of you in making a fair schedule. Your intent is not to "win" and for him to "lose", but to come up with something that WILL BE less than perfect, but will be fair. The goal is each of you have the same level of burden in inconvenience, but also the same level of joy at having the girls. Perhaps spelling it out to him that you're not after the jugular will help. 2) The documents could outline (IF you are okay with it) that you cannot move without consulting him. This is tricky. I don't want you legally in a box. But is there language that maybe outlined the next 2 years that you won't move outside of X kilometers away without additional mediation?
I don't think your H deserves to be coddled, but he clearly can't handle these real conversations. I wonder if a bit of calm rational that outlines where he is "winning" might adjust his tone.
Last question: He does not want you to sell the house. Early on in your convo it sounded like he outlined one option being that he owned the house (instead of you). Is this not feasible, not wanted? I'm guessing you wouldn't want it, but just want to be sure it's ruled out.
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In some of the older threads on this forum there were some who would advocate for the rare and well-timed "truth dart". The rare time you really put everything on the line and outline to the WAS exactly what hurt was caused. It goes against the current diatribe of "no R talks - EVER" that most speak of currently.
Anyway, what if this is the time you send a little dart? I know you both are prideful so this may not be what you'd like to do - you may be past it. But you could consider, "H, I am trying to work out logistics of a difficult but realistic decision. I was not the one who chose to leave the M. You did. And we have been living as if "almost married" for 2 years. It has been comfortable enough for both of us, but not sustainable. Me buying a new home is not something I want to do or something I am doing to spite or be vengeful of you, I am merely addressing the reality I was given by your decision. And with that, we need to work out these details regarding (Sale of house/custody/child support) as best we can. I am asking for you to work through this with me with mutual respect. Unless your heart or decisions have changed over the past two years, I do not see an alternate way forward".
Post often, FS. If you're up for it. I was thinking of you today.