Or we could just have the float plane cost be abstracted into the cost of the whole ship like it is currently and not have to make an entire production line to produce a handful of planes.
The cost is abstracted into the seaplane module, not the whole ship. Personally, I wouldn't be that keen on having to produce floatplanes for BBs and search cruisers, since it's such a small number. You'd put one mil on it and forget about it, like scout planes or transports if you use those. Just give the the planes to the ships "for free", as is done currently, no accounting necessary.
But given that there's a plane designer, I don't see a huge difficulty or harm in using those stats to calculate the search value for that seaplane module. No airfield icon on the non-carriers, no mission assignment. The module just bumps the stats of the ship much as it does now. If the small differences you can conjure by fiddling with the floatplane design aren't worth it to you, great; just stick with the default design. Come up with a new design, and it bumps the stats of new floatplane modules. (You could have a floatplane module refit to upgrade if you weren't willing to just bump the value for all floatplane modules everywhere.)
I think we'd want a production line for bigger seaplanes to match scout planes. Use the current air code, and base the planes from ports. Only planes with sea/floatplane modules can base at a port. Make the basing limit small -- say, 10 per port level instead of 100 per airfield level -- and there's no worry they'll somehow dominate land-based air (especially given the disadvantages I expect come with adding the seaplane module in the first place), or even carrier-based air, even if the seaplanes have a naval attack value that comes with a secondary torpedo armament. The ports can then search adjacent sea zones and mount a small naval attack value good for sniping at subs/convoys (as Catalinas were designed to do), small TFs, or just thrown into a big battle for a few extra attacks because you expect that every plane will do their duty for England.