How much energy do you need to successfully auto rank off the chain, with Neural and RSL? Is it around 12x your rank? Currently at rank 366.

Energy required to autorank off a mission = (40R + 20)/([Exp ratio of mission] x 1.265)

Where R = current rank, (40R + 20) being the experience requirement of that rank.

You need 14.660 to go from 366 to 367. With a neural and RSL you will be getting 778 exp per click. That comes out to 19 clicks at 245 energy per click which equals 4,655 energy. That is 12.72x your current rank. The next 3 parts of the mission have a higher exp/energy ratio so you won't need as much energy.

1 minor comment here. 4655 energy is enough to start off autoranking off the 1st mission, but not enough to finish all 250 rounds to completion. At the end of the first mission, 13 ranks will be gained using a nerual and RSL, so you would be concerned about energy required to go from rank 378 to 379. 15,140 xp would be 20 clicks at 245 energy per click which equals 4,900 energy.
if you account for putting the 5 Rank Points per rank for ranks 366 - 378 into engineers as you go, you can get through the first 4 missions when starting at 4780 energy at rank 366 (13.06 energy/rank)

As far as the entire chain goes, starting at rank 366, you will end the chain at rank 444 using a neural and RSL all the way through. You need 17,740 xp to get from rank 443 to rank 444 and the final mission in the chain will give 759 xp / click with neural and RSL. That is 24 rounds to rank from 443 to 444 which would equal 6600 energy. Also accounting for the 5 Rank Points per rank for ranks 366 - 443, you can get through the whole chain starting at 5830 energy at rank 366 (15.93 energy / rank)