I strongly suspect that if you back off the non-drive-side cup a fair bit, the spindle will turn just fine. Basically, the non-drive-side cup is just there to keep the cartridge BB body from rocking around within the BB shell, leading to chainring wobble and eventually messing up the drive-side BB threads. It doesn't need to be torqued down very hard. Shimano has started making the stabilizing cup out of plastic, which cracks if it's torqued down too much.
Anyway, if you tighten the stabilizing cup too much, it will compress the bearings and lead to bearing resistance. Here's how I explained it in another thread
(which was about square-taper BB's but the dynamic here is the same):
Originally Posted by timcupery
My experience with cartridge BB's has mainly been with steel or aluminum stabilizing cups on the non-drive-side, although I've built one bike
for a friend with a Shimano BB with plastic stabilizing cup, and also a plastic cup on the bottom bracket on my Centurion
The non-drive-side cup is there mainly to keep the non-drive-side of the cartridge unit from moving around inside the BB shell (which would stress the shell, and the BB body). It also functions a bit to increase preload on the drive-side threads.
But if you tighten the non-drive-side cup too tight, the bearings get compressed and don't turn as freely. You can easily try this by tightening the non-drive-side cup (if you've got a metal one) and seeing how the spindle becomes more difficult to turn between thumb and forefinger.
So what I've done is tighten the drive-side, which is always steel or aluminum threads, quite tight (I don't have a torque wrench, but it's probably near spec). I'll snug up the non-drive-side a bit, but keep testing the spindle between thumb and forefinger to see if I've increased bearing resistance. If I have, then I'll back off on the non-drive-side cup.
With the plastic cup, I just go by feel and don't tighten it to where I'd worry that it would crack or threads would strip.
It may be worth your reading through the entirety of the other thread.