How do I remove a flush mounted ceiling light fixture dome?
I had a very similar light, and the key was this:

Any upward pressure exerted from grabbing the glass created friction that made it not turn.  

You want to touch as lightly as you can, near the edges, and try to apply rotational energy only, with no pushing up. 

Unfortunately, that is roughly impossible to do if you're reaching so high that you need to press up on the glass to maintain balance. So, you still need to solve the problem of getting higher (and you should NOT climb on anything placed on a bed unless it's highly stable).  But if you can get a bit closer, try less pressure, rather than more. 

It feels very counter intuitive, but it helped me in a similar circumstance.

One other thing that may help, suggested by none other than @Joel Spolsky, was to try using rubber or latex gloves to allow you to produce more friction without more pressure.