Time to play the game to win.

If BM has this kid spying and documenting things in your home.... start salting the house with things like photo shopped bills of sale for expensive jewelry, cars, expensive vacations, boarding school for your kids, etc....

When BM blows her stack you can honestly say that none of those things have been purchased or even considered.  If she brings it up in court, tell the truth. None of things have ever been considered and no business of any kind has ever been done with the businesses she is bringing up.  Of course, they do not exist. I doubt BM will even check.  She will run with what you feed her.

Have fun.  Idiots are extremely fun to play.

So is your H going to court to get more time with these nasty kids or just to spend less money on them? I hope for your sake it's the latter. I wouldn't be at all happy with his kids taking pictures of things in your house and making fun of you to BM.

I would tell H to deal with his kid and shut down the disrespect - even if he's worried about how it will look in court.

DH should deal with it by taking away the phone. The logical consequence for misusing the phone is SS does not get to use the phone for a period of time. I realize both SS and BM will loudly complain, but your house - your rules.

We have the rule with SD and SS that they cannot take pictures with any device outside of their bedrooms. We had to make the rule because even though we bought them perfectly nice, new, brand name tablets to use here with full on parental controls, they were brining devices from mommy's house so they didn't have to follow our rules and we couldn't lock the devices (I did however block those devices on our WiFi router so she had to pay for SIM cards!). I don't know where those pictures may end up and our house is none of BM's business. So fine, you want a picture of your messy bedroom? No problem. But BM doesn't get to size up what laptop I have, what tv we have, etc.

Ours act as little spies as well though so the idea of putting out brochures for new cars, real estate, nice jewelry-all of that is good for the little spies to report back. Someone also recommended I leave a pregnancy test in the bathroom, that one was gold!! I haven't gone that far yet, but I have left out a few brochures for new homes near my parents place. Waiting for those emails to start coming in accusing us of moving or buying property and not giving BM more $$, lol!

Well, if you can't tell him you saw the texts, then you can't do anything about it really, can you?  Sounds like a typical alienated kid. Beware that using his texts in court doesn't get you in trouble - it can because you are monitoring his conversation with his mother.  Also, we were told texts are useless because anyone can pick up a person's phone and send a text that appears to be from them.

Catch him red-handed taking photos and confiscate the phone for violating your privacy. Thereafter, take it off him every time he arrives and give it back when he leaves. He has absolutely no reason to be taking photos of your home.

As for him knowing everything, respond every single time with "SS dear, nobody loves a smartarse" and be sure to do it in a condescending tone and don't forget the eye roll.

Remind me  - how old are your DH's kids?  And do you actually have a court date and an end in sight to the custody fight?

Honestly if it's just the affirmations he is just taking photos of I would let it go. He is probably making fun of them because that kind of positive self talk and praise is out of the norm for him at BMs so he makes fun of it. For this part I would start choosing affirmations that would make him reconsider his own actions. Like avoiding gossip, malicious talk and humility. If he is taking pictures of it, then he is reading it. And BM will consider whatever you do as a threat no matter how simple it is (like affirmation notes on the frig) so she will minimize to convince him it isn't something he should like. Let her have it...In fact a little note about avoiding malicious people and their effects on their peers would be my first note. 

As for the racism.....thats really a big deal. She is turning him into a racist. The best response to his "All lives matter" comeback would be to ask him "who said they don't?" and "yes, and when one of them are wrongfully murdered or injured we ask for change to help stop it from happening again so why cant we do that for everyone else who isnt a cop?" I think you need to figure out if your SS is an indoctrined racist already or if he is simply repeating what he is hearing from BM and his peers without actual hate for minorities. When he says racist things I would ask him what makes him feel that way or did you hear that from some else and thats why youre repeating it? As a teacher I cannot tell you how many students (and I taught High School so they were old enough to be aware of their words) repeated dumb shit from their parents not realizing what they were saying was racist and minimizing to miniorities. So speak on it. If he says something thats racist say it to him Calling someone a racist is identifying their words to a sterotype and he will either accept the identifyer or reject it and therefore avoid saying those things to avoid the label. Education (like print out actual articles and facts) and KIND confrontation coupled with patience in explaining and listening is the only way to combat children who are being indoctrined by racist parents. You won't change him by attacking his (developing) character but you can make him think about things. Letting them connect the dots and drawing their own conclusions help solidify their believes as their own so BM can't come back and try to sway him back. 

I think your children could use some tools to deal with potentially racist family members as well. Books, a therpaist, something.