I am considering making a stand from an old rim so I don't have to work with
this in my lap, and to be able to disassemble the saw, work on it, clean it,
etc.

My uncle was logger in the woods of NH, since retired. His
saw sharpenning stand was a maybe 2' long log, cut to
about a 1' square.

He hollowed the center to make it lighter, so that it looked
like 2 solid ends held apart by 4 maybe 2x2 or 3x3 corner
posts. One of the sections he cut out of the middle was
cleaned up and also hollowed out (maybe 15" tall?) but
with only two posts, to be used a stool, and was stored back
inside. Threw the whole thing in the chain tray of the truck.

Might be a little awkward for what you want, but for him, the
materials were what was at hand. I know he also used
stumps when away from the truck.

My personal take on it is anything that will put it at a more
comfortable height for you is probably a good idea.
--Glenn Lyford