Thats a right mess!
Frankly, I think you are putting yourselves, the bank and the solicitor under way too much pressure to complete - and all for the sake of a discount. Why don't you just tell the builder that it will be a few days later than expected but that you want him to agree to extend the discount?
It is an absolute pain in the butt trying to do this after the event after everyone has been completely stressed out on an already too tight deadline with nobody having that much interest in completing the changeover.
If I was the solicitor, I think I would just refuse to do it unless you were willing to pay a double fee for the whole thing. Even then, I would be insisting that you would have to do an awful lot of the chasing to get the thing done.
And you are on risk as a non owner until your name is on the deeds.
mf