More houses will always come up, and your money will work a lot harder in basically any other investment, with less stress.
But if you're that close to FIRE anyway, it might not make much difference and give you some sort of peace of mind/happy anticipation - which is a worthwhile thing to spend money on.
I'd sit down and figure out what you'd make in the market, and then assume you'll break even on the house including the mortgage paydown on a 30 year note - so your cost will just be the expected returns the DP would earn. If it's worth that amount (of extra money and hence working time before FIRE) to you to have this plan/house in place, then pull the trigger. If not, don't.
We did something similar, and bought a house we liked in a LCOL mountain town that basically breaks even when all is said and done. In our case that was just as a bailout option if we decide to leave our HCOL area. I wanted to know that if my business did poorly or I was getting super stressed out, we could simply sell our house, move into our other place, and be instantly FIRE. There is some significant cost associated with that since the house is a crappy rental, but it's worth it to me for the peace of mind.
-Walt