Hello,
Trying to get bridging without a contracted sale of your existing home will be near impossible (it's known as "open ended bridging"), particularly to fund a new build where there will be development risk until the construction phase is complete and the new house has been fully signed off.
Your issues with your three buy to lets probably reduce the chances even further tbh.
Assuming you intend to clear the cost of building the new house (€200k) from the sale proceeds of your current home, would there be any additional funds left over and if so, what do you propose to do with this money ? ... if there was a surplus and you were prepared to put it against the debt on the BTLs then you might have a slim chance that the lender behind the investment properties might do the open ended bridging for you, but I'd think it would still be very slim (and also, it won't be cheap !).
I think you might be wise to consider selling your own house first an renting somewhere for a while, until the new house is built and ready for you and your family to move into. This way, you remove the unnecessary risk of not being able to sell your house exactly when you are ready to move to the new house and you don't have to pay a high interest rate for open ended bridging (assuming you can get it approved by someone in the first place).