Single speed chainline
I would surely go by changing the bottom-bracket for a longer one (be it changing the spindle as you said, or installing a sealed Shimano-type one).

If you don't want too much to keep your current bottom bracket with separate spindle and cups, the sealed one is much more reliable, long-lasting, play-free and water-resistant.

To find out the correct size is not a trivial task, I think you should go talk to some friendly and experient bike mechanic, since there are some subtle factors involved in bottom-bracket crankset interaction, that determine the final position of the chainring.

If your spindle looks long, get a shorter one and move the ring to the outer side of crank spider. If your crank is already very close to the frame, find a longer bottom bracket.

OR

You could play with the wheel/cog position,sometimes there are washers and nuts around the bearings that can be moved, swapped, changed for thinner/thicker ones, etc.
I have a fixie with a very cheap rear hub, and there is a lot of dishing, but I don't care, I've ridden dished wheels my whole life and they worked. It is not prettiest, but it is a minimal problem compared to twisted chainline on fixies/singles.

Hope it helps!