Haven't stayed at CBR, and have no interest in staying in a pirate room, so take this part with a huge rock of salt. (I realize you may have kids, and if that is something they're really excited by, that's a point in CBR's favor. And I'm giving my thoughts as a childless adult.)
I would not pay *more* to stay there during heavy construction... which should still be well under way during your trip dates. You can read about its progress in the CBR thread,
on Disney's website, or on any number of Disney bloggers' websites complete with photos. I'm not sure what state the construction would be in at that point, as far as the dining/shopping/lobby goes. They have a temporary dining setup at the moment. Depending on how you eat, though, that might not be a big deal. I prefer to eat away from the resort for the most part, so the current dining setup probably wouldn't bother me.
So that's one amenity. I guess the others would be pool, theme, and possibly bus service?
Pool is somewhat negligible to me. I love swimming and swim almost everyday during trips, but am not really convinced that the pools at moderates are always "better" because of the slides. I liked that the Pop computer pool was quiet and uncrowded every time I swam.
Theme is totally up to you. I did not think I would like the theme at Pop, and was surprised that I love it.
Can't speak for CBR's bus service, but was happy to find that Pop bus service was frequent, efficient, and often empty (except for at park closing!). Another "perk" to Pop is that AofA is right over the bridge. If you don't mind a short walk, you can always hop on a bus headed for AofA (like if you're leaving a park and it arrives before a Pop bus) then walk back to Pop. Their food court offerings are a little nicer than Pop, with some more variety.
I also love the cheery, clean, sleek, functional, modern, cozy IKEA model home (meant in the best possible way) feeling of the refurbished Pop rooms.

I think they're an excellent value for the price. Not sure I'd feel the same way if I stayed in one of the older rooms, which is definitely a risk until construction finishes. Although I'd hope by late February they'd have all of 70s complete in addition to 90s/80s. That would tip the odds in your favor of getting a refurbished room.
RE: Pop, you said that your dates aren't available.
Have you tried a room-only discount? Or do you need the package?
Bottom line: If there's nothing available Pop-wise, I guess I would hedge my bets by booking the CBR package for now, keeping tabs on the construction updates, and frequently looking for an opening to downgrade out of a pirate room. While also checking for Pop (or a different moderate comparable price-wise?) to open up.
Someone else with recent experience at CBR, please reply. I am totally biased toward Pop after a great first time there in September!