Nice idea.

In a tour of Dr. Phil Mason's lair, he we see cameras indoors at windows, covered with a box and black cloth. This Video includes three time-lapse views in different directions.

The weatherproofing enclosure is the house!

Since the point is to show the analemma which has the sun move left and right, you do want to expose at 24 hour intervals, not adjust for true noon. But, you can hedge your bets and take a group of pictures every 24 hours, not just 1. Say, 30 seconds apart for 5 minutes on either side of noon.

An off the shelf intervalometer worthy of the name (as opposed to just a timer cable release) should have two levels of interval. The generic Chinese imports are $10 as I recall. Get two, so you can immediatly replace if it dies.

A dedicated Linux server is overkill. You could just use an existing computer if you needed more complex control, unless it was physically isolated. But a custom intervalometer that has the specific custom schedule would be an Arduino, not a Raspberry Pi, I would think. Maybe you want 3 levels of looping, geometric time sequence instead of linear around the target time, sequences at different times in the same day, etc.

You'll want to check on it, probably maintain it, over the year, so you need some way to reproduce the position when you put the camera back. Sight-lines to reproduce the view would be most robust.

If you make a movie, you can see the changing seasons in the view, Rod Taylor style, with the sun moving back and forth. You can make a video showing several presentations of the data, first just the frames, then again leaving traces in the sky, looping faster and faster so the sun becomes a blur.

What about cloudy days, rain, etc?

Another thought: the exposure for ground, sky color, and sun would be different! Set the camera to a bracketing burst from one shutter press. The remote shutter timer can't change the exposure so the camera needs to handle this 3rd loop level, itself.

Ah, but cameras can be controlled in detail now over a cable, if that cable is USB. You can have your regular computer send complex commands via WiFi, and also download each day's raw files without you having to touch the camera.