Push Reel Mowers just don't cut it.

Ok I learned a lot about hybrid Bermuda grass as I had my new yard sodded with it and it was overseeded with unkown type of Rye grass from a local sod farm. The 920 sq ft of sod was installed Feb and the cool season Rye grass established quickly. I started mowing with a Great States push mover in mid March. The Bermuda grass had not yet woke up as soil temps are low 60s. My desert location is near Las Vegas NV. The heat really turns up in July and Aug and the Bermuda grass finally woke up. The rye grass doesn't like the scorching heat and yielded to the heat loving Bermuda. The push mower set to the lowest cut height at 1.35 inches. I took off the roller bar and able to cut to the lowest possible at 1.25 inches. Looking good so far. Fast forward to Oct. All the golf courses here close for a couple weeks to overseed for the cool season, as Bermuda goes dormant. Mine still mostly green, but slowing down and does have a couple of thinner spots. I also decided to overseed. Did the dethatching (scarified)with a Sun Joe and researched for cool season overseed grass. Found Penncross Creeping Brent was a choice that can live with the Bermuda. It is a very fine bladed grass used on golf course greens. It wants to be cut short. I overseeded in Nov weeks behind the golf courses. Wasn't long and Brent spouted. Dec,Jan,Feb nice green look vs the now brown, not golden, Bermuda. Mowed couple times. Love the beautiful green soft carpet lawn the Brent provides. But now here is my problem. The Penncross Brent wants to be cut short, under 1/2 inch or less!!!. I have looked but cannot find a push reel mover that can mow to a greens level height. Lowest is a fiskars reel and thats barely under 1 inch. I'm wanting to stay gas free and eyeballed a couple of battery powered greens keeper mowers but they are serveral thousands of dollars and cant justify for 920sqft of lawn. Dont think the Sun Joe corded cylinder mower is the right answer either. So my question, Is there a manual mower out there that can mow to a greens height? Any direction on this would be greatly appreciated. Thanks