"(where we would have to apply for a 90% LTV on the 500K as opposed to a mortgage to meet the 250K diff)."
Is it not all the same difference?
You will be looking for a mortgage of 250K to buy a 500K house.
It is only a "property swap" - trendy term for two separate transactions that need to happen at the same time to work- if there is no lender involved. Stamp duty is still payable and you will each need a solicitor to act for you.
Once there is a lender involved, all the usual criteria apply and all negotiations need to be on the basis that you will have the funding for the difference.
So, you need to sell this idea to your lender.
mf