I'm pretty sure getting all the upgrades for the Crows Nest will cost diamonds ...13 full trips to gather them all is 260 spaces...the math just won't bear out. That's OK though..it's a freaking monster of an fp producer and INNO shouldn't just give them away. I may spend the bucks for diamonds to get a fully upgraded one simply for the aid to fp to tile ratio. Meanwhile, ship upgrades on the wheel is an excellent prize and kudos to INNO for not making past event prizes obsolete....
I'm not sure that you math & assumptions are correct.
Here's what I tally...
CN is 8 levels in total. If you complete the entire questline, you get CN levels 1 thru 3. The remaining 5 levels must come from the ship map/compass - winning another CN level for the 1st, 3rd, 5th, 7th, & 9th complete trips.
Each doubloon used will provide 1 compass for the ship map. Each map trip is 20 spaces that your ship must move via the compass, with each compass giving you either 1, 2 or 3 spaces. So far in all my worlds, I've been averaging 10 compass spins - average of 2 spaces moving each time - to complete a trip (have ranged between 8-12 compasses depending on randomness, but average amongst all is 10 for me).
While I've no proof 10 is actual average (too small sample size, etc), for discussion's sake if I assume it is... We need to make 9 trips to get remaining 5 CN levels, each trip averaging 10 doubloons to complete -- for total of approximately 90 doubloons needed overall.
Doubloons we can get:
3 - to start the game
20 - for the two questline milestones of 10 each (the other 3 milestones being the 1st 3 CN levels)
56 - one per completion of each quest except last quest gives two, with 55 total quests
21 - daily freebie for logging in, one per day for 21 days of event
Total = 100
... which doesn't include any doubloons found in incidents or the 2 freebies you can collect via Inno's event annoucement email sent to your registered email address (note: if you play multiple worlds, make sure you log out first, so you're sure to open the world you want them in when you click on the email's free doubloons button).
It's a tight fit, with not much extra room for very bad luck with compass movements on ship map. But it does seem possible to get fully leveled CN without diamonds.