This is where you need specialist advice. I think there are two separate issues here.
One is dealing with the building society and the other, separate, issue is the S106 agreement.
The S106 issue is probably easier to deal with. Does this just involve a payment of money or are there other requirements as well?
If it's just money then it depends how much they're asking for. I'm aware of one council where the S106 payment for a three bed home is around £2,500. In contrast, where I live the CIL payment for a 100 sq metre new home would be between £14,000 and £20,000.
If the S106 agreement requires a lot of money then you might wish to approach the council and see if they have a self build exemption for S106 (most councils do). If this is the case then you sign the S106 agreement and then apply for the exemption afterwards. But get advice specific to your own council.
However, this is only for self builds so you would need to move into the new property yourself and use that as your main residence, usually for at least three years, while letting out your current property.
If you don't move into the new property yourself then you can't claim the self build exemption.
I think the issue with the building society is a different matter.
"maybe it could be possible to take the land beneath the new house out of the mortgage as we have sufficient equity to do this..."
This would be a straightforward thing to do - Form TP1 (transfer of part of registered property) would create a separate title and is a standard thing that any experienced property solicitor can do.
"...but not sure if this still leaves an issue with the new house being built attached to the existing house."
It just creates a party wall and that's fine as long as the owner of the existing house (you) agrees.
As to what the building society would say, I don't know, but I imagine they would take the same approach ie revalue the property as a mid-terrace with a small(-er) garden.