Keep in mind that with a fixed gear, you'll have to pedal down the hills as well as up. It's a strange feeling, definitely different than bombing big hills with a freewheel setup. You may want to put some thought into creative options:

A flip-flop hub so you can run a fixed cog on one side and a freewheel on the other. Going down big hills fixed is a lot of work, and it's nice to have options. Since you're buying a new rear wheel, I would definitely consider this for flexibility. Beats being stuck fixed if you don't love it.

A Sturmey internally-geared hub is another route you could go. These really aren't that expensive, and the S3X is actually a three-speed fixed hub, so you'd have a low fixed gear for climbing and a high one for descending. You can also keep your nice singlespeed chainline, and they require significantly less maintenance than a derailleured bike.