Hello JuniperJuna: I'm sorry you have not received any replies to this post. I don't know as I have much of anything useful to suggest here. But I noticed that no one had replied to your post. So I thought I would.
From what you wrote, it sounds like there is no simple solution to this problem. It seems to me this is a situation best addressed with the help of a skilled family therapist. Ideally you, your partner, & his son would participate. But if that is not possible, it may still be of benefit if you go alone.
The only other thing that occurs to me is to establish a schedule whereby your partner's son stays in each home for specific periods of time... weekly, monthly, etc. And he must stay at the home he's scheduled to be at.
As far as the problems he brings up, perhaps he should see a counselor or therapist of his own with whom he can work. If the therapist determines that there is reason to think there is actual abuse occurring, it would presumably be the therapist's job to report this to the appropriate authorities. I don't know how this works where you live. But where I live, therapists are considered to be "mandated reporters". They are required by law to report suspected abuse.
Of course, this could backfire. Your partner's son could well make up stories about you & your partner as well as with regard to his mother. It would be important to have things set up so that the therapist can check in with your partner & his son's mother as well. This, of course, assumes at least some minimal level of cooperation on the part of the mother, which may not be there.
Beyond this, the other thing that occurs to me is to simply tell your partner's son that he must work out the difficulties he has with his mother... with his mother. And, if his stories suggest there is abuse of some kind occurring, tell him that you will henceforth contact the appropriate authorities. So it is important that he be truthful. Then follow though as necessary. If he is telling tales, he needs to learn that there are consequences to his stories. I don't know how else you break this cycle.
Perhaps other members, here on PC, will yet reply with better suggestions. I wish you well...