"Oh, Suzie won't feel left out! I ask her to do one-on-one things too, just like you do. I just want a chance to go to this event with just the two of us". Then ensure you offer a similar invitation to Suzie.
I agree with this. Make it clear to Jane that you also plan to spend some one-on-one time with Suzie also.
If Jane (and/or Suzie) comes back with "But it's more fun when the three of us all hang out together!" I don't think there's anything wrong with gently pointing out that sometimes Jane and Suzie do stuff without you.
And then they gently might point out that they are closer friends with each other than either one is with the OP. Does the OP even want to have one on one time with Suzie? I didn't get that vibe by the description of what she has in common with Jane vs. Suzie. If not, I think it is going overboard to have her create a plan that the OP and likely Suzie really have no interest for the sole reason of removing Jane's objection, which is likely simply polite excuse anyway.
I don't know, if I made it clear that I prefer to do things as a group with a particular friend and she kept suggesting ways for her to split us up and do things one on one, I'd find it weird/annoying/and likely high maintenance and start to distance myself.
The OP has already made it clear to Jane she wants to make plans without Suzie included. Jane isn't interested for whatever reason. Maybe Suzie has no interest in hanging out with the OP without Jane. Or vice versa. And there could be a variety of reasons for that.
I have some friends I find pleasing in a group but they are too quiet or too difficult to maintain a one -on-one conversation with. Put them in a group of three or more, and those issues are diffused.
Maybe Jane wants to go out and not talk about family or kid stuff the whole time, the things the OP says she has in common with her. Maybe she goes out with the girls to get away from those pressures and talk about other things, and Suzie offers her that outlet. And a combo of the two of them gives her the best of both worlds.
I am just speculating here, but I think you can politely hint and ask for one on one time with a friend when the usual dynamic with YOU and these particular friends is to always see them together. If they balk, it's time to throw in the towel.
What they do together really has nothing to do with it as the OP has said they are closer to each other. All that proves is that they do really enjoy hanging out together. The OP feels more comfortable one on one with friends, I'd suggest she finds other friends who do as well, and accept if she wants to see these friends, they come as a package at this point.