I would tell gf that you are going to do this & then
I'd tell Sun-that you are going to be busy and unable to keep in touch for the next two weeks to deal with some family drama.
Then, I would focus some much deserved extra love and attention on gf (in light of the family bs over her coming out). NO DISCUSSION about SUN or any other drama or needs on your end-just focus on giving her the love and support needed to manage the shock and pain of her parents response. Remind her that this too will pass-and you're doing it together.
At the end of that two weeks, the gf should be a bit more calm and mellowed having gotten some needed extra tlc from you. Also, it will give you some time to contemplate where things stand for you in regards to Sun.
THEN I would sit down with the gf and discuss exactly what it is that she needs to be comfortable with you dating other women and explain to her exactly what it is you are looking for (not using names of the current women or the past).
THEN decide what you are going to do about the Sun situation.
IF there really is a connection and she really is sincerely interested-there's absolutely NO reason she can't manage a 2 week pause so that you can be sure your gf is ok after her parents bs.
Being in lesbian relationships and in poly dynamics-we are all more than capable of understanding how devastating that is.
To add that to having depression issues, whilst she may not BE a suicide risk-the risks rise with such parental lack of acceptance and we all know-that the first thing we need to do when that shit happens is buckle down to support the person who came out in such a painful circumstance.