Yes, you need to protect the cockpit interior.  It does not need as much UV protection as the deck because it won't be exposed as much, but over time it will go cloudy if you don't do something.  On boats with huge open cockpits like your Mill Creek, I have seen painted interiors which look awesome.  On my boats with smaller cockpits, I usually apply about three coats of varnish before adding the seat.

   I'm in the same boat so to speak - just finishing a Mill Creek 16.5. I varnished the cockpit before I put the deck on as it was easier to reach then. Thought about white paint but I can always do that later, right now it's easier to varnish everything. The cockpit area will be exposed to UV so it does need some kind of protection. I put the first coat of varnish on the deck this morning and touched up a few areas in the cockpit at the time - the hull already has 4 coats of varnish on it.  

Launch party next week - 

BTW speaking of cockpit protection I found an outfit that makes a tandem cockpit cover that looks like it will work - I have one on order and will report results when it gets here - I'll try it out on a 1,000 mile camping trip I have scheduled - the MC will be on a roof rack with the cover on it. Here's the link - not an endorsement as I haven't got it yet - 

https://www.storeyourboard.com/tandem-kayak-cockpit-cover-universal-kayak-storage-drape/