@BB if M/s Sue, Grabbit & Runne (Solicitors) are involved it will cost money.
There sounds to me to be some basis for an agreement, particularly if both are sensible about access to their son which appears to be the case.
The father would have maintenance obligations which could become expensive if the mother had to move out.
It would seem to me that if you laid the out the financial situation on a number of various options then it might become clearer to both parties.
Are you friendly with both parties or one?
The unmarried father is a constitutional scarecrow and has relatively little rights but he has obligations. Not to put too much legals on it but the most he has are guardianship right and consultation rights - but are qualified.
The property is a different matter. If there is FA equity today and one or other is prepare to pay the mortgage - would that not suggest that this needs to be broached? As a Tracker Mortgage guru yourself - whatever interest rate is being paid could be crucial. Would the Bank ever allow one party off their obligations .. depends.
If there is no chance of splitting the house - folk do this ! - then it seems that the renter would have a bigger cost. I am figuring that the male party may be overlooking the maintenance problem. This would arise - as it seems the Mother will have a cost either way.
I would - assuming no adverse other facts - that the father has about a NIL chance of being given custody. For the avoidance of doubt zilch, zero, none - unless he could establish that it was in the child's interest. It could be an interesting legal battle, cause endless stress, deplete funds and end up with a judge telling him to go for mediation.
Actually why don't you mediate? You could get a Counsel's opinion for under €1k (you are an accountant and could potentially hire a Barrister directly) and then you would know the legal landscape.