I am a bit late to the party, but i found the challenge fun, so I spent quite a bit of time with it. You can quite easily do all challenges with almost no losses from your side, except for impossible.
I am here to show you my impossible variant with a total cost of 198.720. You can go cheaper by 5-10k for sure, but it might lead to a bias that the battle becomes a dice role. With this composition you win about 90-95% of battles , weather off, night/day + month don't matter.
We will build:
1) roach destroyers
80 of them.
2) Heavy cruisers as capitals. 15 will suffice. You can also win reliably with 14. Less can become a dice roll.
3) And of course carriers! as damage dealers. 5 of these will kill the royal Navy, all 5 to your death stack. 4 first full with naval bombers, the fifth, which will receive a penalty, full with fighters. Penalty doesn't work for fighters so it doesn't matter.
You can probably take all guns off and will work for even cheaper buck, but carriers must survive to kill the whole royal Navy, so it might be risky.
SOOOOO. We have 5 CV with 320 NAV and 80 F2, 15 CA and 80 DD.
What can such a small force do, you would ask?
BAM.
BAM.
BAM.
BAM. BAM.
I am actually surprised that someone could get good enough results with BC or BB. They are very underperforming, but could be ok, probably, if you get a good HA or LA number to one shot things and still have good speed. In my opinion, BC/BB can't beat a CA fleet with the same cost. The SHBB or rather SHBB/CA mix, however, can try. Due to the hp and possible high speed.
I can beat a Royal navy even cheaper as Japan or USSR (6 CV with fighter kamikaze), or by using designers (those with reduced cost).
There is still room for experiments, but meta is meta. LA CA to kill screens, CVs with NAVs to kill everything else. BASED.
Those who want to experiment, here are some saves.
I am a believer of an idea of naked SHBB to tank damage:
I did some testing with them and that it is what a fleet 5 CV, 3 SHBB naked, 10 CA of previously shown design and 80 DD can do (212k IC total):
However, they don't have enough LA to reliably win against royal navi (Wr around 70-80, a bit less than by pure CA).
Summary:
1) CA - most cost effective capital, very niche uses for SHBB, probably just garbage BC/BB
2)
CV op, you can abuse the way penalty works. CV op, if you switch the weather off -> three sorties, NAVs go brrrrrr
3) You need 5 CV, 320 NAV, 80 F2, 15 CA, 80 DD to beat the Royal Navi (198 720 IC)
P.s. I apologize, I just realized I could do spoilers ;D. For those who want to try the things out, I would recommend to experiment with one shot/least shot values for attack and the reverse for your own hp.
Edit1. Typos
Edit2. I realized with 4 NAV carriers you get 320 not 240 NAVs (I am a computational biologist from a job, so it makes it even more ridiculous, but happens
) I corrected it everywhere.