Assuming the J-Pac is correctly connected...
It's rare for a J-Pac not to sync properly, but not impossible. You could try bypassing it and combining sync for yourself to test whether the J-Pac or the chassis is the problem. It’s probably just the pair together. The issue with leaving out the j-Pac in the long term is that you lose both 15kHz protection and video amplification. The former can be replaced with Atom-15, and for the latter you might try either turning up the video pots on the chassis or using a THS7316 or similar IC.
To combine sync for yourself you would get a VGA breakout, or build one. You feed the colours directly into the chassis – if you can’t see anything, turn up the screen pot on the flyback and all the video pots. Each sync line can go into a 500R resistor and then a 1k potentiometer, then combine into a third 1k potentiometer set to zero at first. Dial the first two pots around and see what you can get. If you get a stable image, replace the pots with resistors. Once stable, dial up the third pot as far as you can – if it tops out, add a 1k resistor and do it again.
This is pretty ghetto, but every Arcade, TV or PVM chassis I’ve ever tried it on has worked. If you’re not happy with it, something like an extron 190f might work as well. But if the above doesn’t sync first, I doubt the extron will. And you may still need video amplification of some sort – you shouldn’t need to amplify sync though.
https://shmups.system11.org/viewtopic.php?f=6&t=62265