You are in a shitty, messy situation. Have helped others in the same sort of place so this is what I suggest you do:
Get hold of PTS/SP materials (hard copy if you can) and get a good lawyer who is happy to take you on. One that's aware that the fight will be against the kult only until the kult decide that it's NOT in the kult's interest to have their dirty laundry aired in a courtroom setting. Kult will then pressure your ex to give in to keep kult out of courtroom.
Get hold of Kult School materials. Get hold of anything in the public domain from the kult about CCHR, WTH, NarCONon etc., and get the real facts that blow the kult PR crap out of the water.
Where on the Bridge is your ex-partner? Are they on a regular study time? Get the relevant references that show how much time the kult expects them on study/in session AND the prices for said "Bridge to Total Freedom" steps. If ex is OT V or above, get the data about 6 month checks: how much time they take, how much money they take, etc..
Get Hubbard's narky words about "wog justice" and "wog laws" and keep them handy. Your legal team need to know this stuff too and have it at their fingertips. Get a few ex. kult members to help - preferably local to you. Having an ex or 2 (preferably trained and processed) willing to sit in a courtroom with your legal team could be a wonderful asset.
In short, make this about keeping YOUR child out of the kult. And why you HAVE to do that for the sake of said child. And how the other parent cannot in good faith give the child time and resources because the kult will be demanding them and $cientology IS the greatest good per kult teachings and all else MUST be subservient to kult. You don't have to keep the kid away from the parent, just the kult.
It's a battle that's not gonna be unicorns farting rainbows of glitter and sparkles. But as a former member of this board used to say, the best way is to get $cientologists practising $cientology in front of witnesses. Then peeps sit up and will take notice.
Hopefully you can work this out with the ex. without having to go to the trenches. But be very aware of what Tom Cruise did to Nichole Kidman and their shared kids - that's what you can't let happen for your sake as well as your kid's.