Cannot give you exact help but to say I've had the similar problems, with the left crank once bolted up rubs on the near side NDE, also the same with the NDE side retainer ring screwing in all the way. There was a similar posting on here recently, & I dug out 6 NDE BB retainers cups they all had the same OD thread but were all different length, & different I.D. as well I posted the photos but since deleted them from my photo files.

As you know doubt know going from 113 to 116 will only give you 1.5 mm each side, I recently built up a BB assembly having took out a 113.mm axle, & using a different crank had to install a 126.mm axle so that the small chain ring did not hit the chain stay ??

I think you have a severs case of trial & error, sorry cannot be much more help, go for longer axle.