I would still prolong it, just to be a burr in their oversized asses! My comments are specific, but may also provide some thoughts for others. It's a case of playing just within the boundaries of their rules, without playing by their rules (if you know what I mean?).
Firstly, I would insist they keep Gina out of this. At every stage where you do have contact with them, make it plain that none of it was her fault. Maybe it was even against her express wishes. She'd have to play this line, too, when they inevitably contact her. She could possibly maintain this holding position for years, if she really wanted to. At least enough to help her family see that you haven't both sprouted extra heads, or something.
This would undermine their right to hold a JC for her, though they might send around a couple of Elders to have an investigative shepherding call ().
Also, if that child ever makes their own way out and eventually hooks up with Auntie Gina, she can say she was there for them even if the family put up barriers in the intervening years - there's a long term view to be had, if you want to look at it that way.
In your own case, you could ask for a postponement, but that would probably only buy you a week or two.
When they inevitably DF you (because you really should meet, if you want to avoid them cutting things short in your absence), you can wait up to a week before asking for an appeal (don't let this time expire).
They then have to get an appeal committee together, which potentially could take another couple of weeks, especially if the dates they first propose are inconvenient.
In the case of dates, it sometimes pays to be proactive and actually suggest dates to THEM, sometime about three weeks away. If you make it more than a month they will know you're taking the pissant.
Whatever you decide will take guts. Having that final "death-knell" is both a release and a personal blow, which continues with the shunning.
Don't be surprised if one or both of you go through the trauma of grief-like symptoms.
(((group hug)))
LT, of the "ex-Elder" class.