Touchy family situation..Invite non-divorced future father in law's girlfriend?

Touchy situation for me to figure out--
FI's parents are not legally divorced but have been living seperate for about three years. FFIL has had a girlfriend for the last two years, and FMIL is single.

I personally think that it would be fine to invite the g/f of the FFIL but when she comes up in conversation with my FMIL (whom I adore completely) gets upset and resentful. It's not that she wants to be with him, but she feels like they are still married and he is an adulterer. She also doesn't want to have anything to do with the new g/f.
At FI's sister's wedding about a year and a half ago, the g/f was not invited.
My FI and I have had dinners with his dad at his g/f's house, spent holidays, etc.  It feels rude to me to not acknowledge the relationship, but at the same time my FMIL's comfort is important. This would be the first formal situation where the g/f, FFIL and FMIL would have to interact.
Both FMIL and FFIL have large families which will be invited, on FFIL's side has met the new g/f.
Is it appropriate to invite the new g/f? Or for the sake of my FMIL not invite?
My FFIL is kind of weird, and I wouldn't mind having his g/f be "in charge" of him all night to keep him "appropriate".
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