Given that I'm going to have to use a R foot Pantin, which hand should I use for the Ascension?  I normally use R, but thought that perhaps L would 'balance' the system more?

I do have my Pantin on my left and jammer on the right but as I am right handed I don't think I would like my hand jammer on my left though and pantin on the right as you have it. However, I would not think you need to balance out the system, if you put your other foot in your foot loop, as I cant see how it would make much of a difference.

Makes sense re the foot loop.

If I am RH and use a LH Ascension, my RH is free for doing other stuff.  Advantage or not?

I would use a basic and "cup" both hands behind it. This shares the load of supporting your upper body between both arms.
The same technique can be used on an ascension - ignore the handle.

I agree with Cavermark.  I have a RH pantin and a basic, but mostly use my RH to operate it.
The 'balance', IMO, is between your right foot with the pantin and left foot in the footloop

I use a right-handed jammer with a left-foot Pantin, and it seems entirely logical to me to use it that way around. My right leg is stronger than my left anyway, and I find it much more useful to use that as the 'driving' leg in the footloop, and (for me) it's more comfortable when resting on the rope. Also my lifeline connection is way over to the right on my D ring, so it's all nicely separated. Now I've learned how to ropewalk properly (at least up vertical walls), I'm going to get a new model Basic soon to see whether there's any substantial difference in speed/efficiency, but as a right-hander I'd still always prefer a left-foot Pantin no matter what.

It doesn't make any difference.

Cap'n Chris said:

It doesn't make any difference.

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Unless you only have one arm.

Personally I use a right-handed Ascension (safety cord on extreme left of central attachment) and right-footed Pantin, but most of the time I have my hands on top of the ascension rather than using the handle.

It's probably slightly easier for me to fit/remove an R ascension with my right hand than an L with my left, though I dare say if I had an L, I'd get used to it to some extent.