Transfer of Shortfall to another Mortgage provider?

My Wife and her sister have a joint mortgage on a property, that they would like to sell. There would be approximately €120K of a shortfall following the sale. BOI have agreed to the sale provided my wife and her sister both take out new separate mortgages with BOI so that the shortfall can be split 50/50 and added to these new mortgages.
We have applied for new mortgages however the rates/terms of the offered mortgages from BOI are not the best option available to us. My question is, are we tied to BOI, or can we take up a mortgage offer from another provider and transfer to shortfall to them if they agree, when we eventually sell the house, or do we have to take up the BOI offer at least until the sale is completed and shortfall added to mortgage and then switch?

Another bank is not going to let you borrow more than the value of the house you are buying and assuming adding the shortfall to the mortgage does that then it's a non runner, your only option is BOI.

If when you sell and add the shortfall to the BOI mortgage that brings the mortgage to near or over the house value then you won't really have any switching options either.

Thanks for the reply Monbretia, we were hoping to build on a site that we own, using a mortgage of €160K and our own savings of €60-70K. The value of the new build house when finished should be in the region of €250-270K, going by similar houses in the area.

Will the other bank allow you use your site value to fulfill the deposit condition
? Then you could use the savings to clear your half of shortfall and not be tied to BOI, you'd end up with a bigger mortgage but that's going to happen anyway when you add on the shortfall.

Its not something I have considered asking, but sounds like a very good idea to investigate further. Thanks for the advice