Is it possible for FAP to prevent you from going on rotation if they believe it is necessary?

The FAP does have the authority (AR 608-18) to intervene to prevent and address issues related to the well-being and safety of military families. So the short answer is yes, their intervention can compel the Garrison Commander and your unit Commander to prevent you from going on rotation. If military law enforcement have been involved and/or there is an impending or pending investigation related to the issue (which is often the case) you can expect that commanders will make decisions to ensure a prompt investigation. It is not FAP's place to say that you will likely be assigned to the Rear Detachment; that is your commander's decision and FAP's authority does not extend to direct command authority (even though FAP falls under provisions of Title 10 of the US Code). Commanders participate in various committees under the FAP and work closely with them. Brigade and Battalion Commanders will be most familiar with the framework of military regulations and policies that the Family Advocacy Program operates within, and FAP's recommendation to them carries A LOT of weight.

FAP can't stop anyone from deploying or PCSing or anything. They make recommendations. I had FAP involved when I told my CO and 1SG about the final time my ex husband assaulted me. Then I got a compassionate reassignment for my daughter and when I was clearing, I had to go to them to clear Behavioral Health because the DV case was open. I went. Told them I was staying with him because he said he'd go to rehab, etc, and they berated me. But signed off. The case transferred when I went to Offutt AFB and an AF DVA contacted me. I did eventually divorce him but having a case in FAP didn't stop me from PCSing.

I guess it depends on why they're involved with you or your family why they'd recommend you don't go.