Temperature for lager fermentation?

My fridge, when set to its 'lowest' temperature setting, stays at about 45-46F.

Do I really need to install an extra thermostat to further control the temperature?

White labs recommend 50-55F for most of their lager strains, so 45-46 would be a bit colder... Other sources recommend 45-55F for lager fermentation.. (BYO - 10 Keys to Great Lager) .. so it seems it would work too?

My plan would be to ferment in the fridge set to low (45-46F), then take the carboy outside the fridge in a cool place (60-65F) for a diacetyl rest for a few days, then transfer in a keg and back in the fridge set to medium (40F) for a month or two of lagering before tapping.

would it work?

you really should follow the recommended temps for the yeast being used if you want the desired results. Can you push the limits and ferment cooler or warmer? Sure. Will it be what you expected? Who knows unless you try. They give you the temps the yeast performs best at for a reason. If you want to push it either way that is up to you.

Perhaps try to use only yeasts rated for the temps you can reasonably handle for now. See how it goes. Good luck.