Rochester appears to be your best bet. Weekend car rentals are usually cheap.

Running one-ways for PHX-LSE with a 4 p.m. Friday departure shows nothing that will get you to LSE before noon Saturday.

Jefferson Lines runs buses between Rochester & LaCrosse, but they don't put their schedules on line. You have to request a quote.

Greyhound has (or at least use to have when I had to take it several years ago) a ~ 1 a.m. departure for Madison that stopped in Tomah for a meal break about 4-5 a.m., but that's barely a few miles closer than Rochester.