Is it Bermuda? I think you’re on the right track, bury it in sand and drag something across it to level it out, you might have to do it more than once, but that will smooth it out. If you want to go low you’re going to have to get a reel mower, the only hard part is the maintenance but you can either find a mower shop or lots of golf course Mechanics will do side work sharpening reels and bedknives.
My only experience with lawn reel mowers is with a Tru Cut back in college, I’ve always had Deere or toro made for golf and sports fields. Since 2010 a ton of golf courses in Georgia have gone from walk mowing their greens to ride mowing, so there are a lot of used walkers out there that could probably be had cheap. They are set up a little differently, they don’t have rubber tires. They have a metal drum, but you can definitely use them on your yard. Jerry Pate is the Toro dealer, and Beard equipment (they just bought Greenville turf and tractor) is the Deere dealer. It might be worth a shot contacting their used equipment departments.
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