Poly is all very new to us and the most important thing is communication, communication, communication...
Coming to it and making the first thing you say "I have to be honest, I have been thinking of splitting up" when you've clearly designated to try to work through things seems from my perspective to be a manipulative tactic.
Yes, communication is important. But communication and honesty don't mean dumping every thought that enters your skull. It means thinking first and balancing openness with respect and empathy. Maybe it means that even if she does feel like she has been thinking of splitting up, it may not be constructive and helpful to say so.
It's important to consider whether what you're going to say is going to move things forwards and help build foundation, trust, and compassion, or whether it's going to move things backwards and destroy those things. In this case, I think beginning conversations with a declaration that she's thinking of splitting up moves things backwards. So, she could keep it to herself. That's not being dishonest or failing to communicate. It's being respectful and compassionate.
I'm not going to rush her on things as I know these are genuine insecurities. Insecurities are not something you can control, she's not consciously saying no to me and J because she's a nasty bitch - quite the opposite!! She feels horrible for saying no and has been beating herself up over it.
It's not like saying no is an irreversible mistake. If she truly feels horrible about it, then all she needs to do to relieve the guilt is say yes. Sure, that means she's also saying yes to some uncomfortable feelings and a need to work on jealousy issues, but how much worse can that be than the status quo?
Who gets to decide that pacifying her insecurity is more important than respecting your autonomy? Who gets to decide that the pain she feels from you dating J is greater than the pain you feel by having your choices taken away?
She can't control her insecurity, but she can control how she behaves in response to it. She can choose to respond by putting limitations on you and controlling how you interact with other people, or she can choose to let you control your own interactions, trusting that you'll do what's right for her by continuing to treat her with love and respect.
If she's really in a place that she can't handle you dating anyone (which is what it's sounding like even though she's found ways to convince probably even herself that this isn't the case) then her insecurity may be too great for her to be involved in polyamorous relationships at this time.
The scariest thing in the world is the unknown. She doesn't know what to expect with you and J. So that freaks her out. But the only way to overcome that fear is to take a risk, let the relationship happen, and see where it goes. Maybe that will be awful, but probably it won't be so bad. And even if it is awful, she's already considering splitting up, so that's just the nail in the coffin. But without even giving you a chance to show her that you're able to date J and still respect S and fulfill her needs in your relationship, she's taking away her opportunity to overcome the fear.