The bank isn't going to allow you to make repairs on a bank owned property prior to the close of escrow (neither would you want to in the even the transaction goes south). On most of the REO's I've sold that have pools, and the pool was empty, closing the deal without the pool being filled and operational has not been an issue. The loan underwriter would be the ultimate decision maker on that issue.
$12,000 to $15,000 will go a long way for pool repairs but you should call a licensed contractor for a better answer. You are going to have an inspection period if your offer is accepted and you'll have time to walk away from the deal if you find the pool repair costs are prohibitive.
Call me if you need help with your offer and inspection. I rebate part of my commission on every transaction and you could save several thousand dollars.
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