Book the AMM-BNA flight now, before it disappears. There's no charge for redeposit more than 30 days prior to departure, so the only downside would be that you'd lock in ~10K Chase points as United points if you end up needing to cancel.

Then, wait, and monitor the prices for the outbound. You have eight months until your travel date; that's plenty of time for some seats to open at a reasonable price.

Also, when you reach Platinum, you will gain access to IN inventory, which is often equal to I but sometimes can be greater. (Those are the booking classes for Saver business class awards).

If you have any flexibility on dates, that will help also. But mostly, don't rush into a cash purchase this far out; you're very unlikely to get the best possible fare. Keep your eyes open and be ready to pounce if you see an outbound award flight.