How does your SD feel about being stuck babysitting a 4yr old all the time without pay? Maybe this is a chance for you to teach SD her worth and help her draw up a babysitting contract for BM to sign with her. It is good for young women to start having expectations of their worth to their employer. 

The only time my SD sees her little half-brother is when she babysits him at my house (without my approval)

Number 1~ How is this your house and you didn’t approve of such an arrangement? How does this even happen?

I've told my DH I didn't approve of 4 yr old boy coming over unless it was an emergency. And I've heard him tell his EX that also but she doesn't listen. She calls my SD asking if she'll watch the boy and of course she loves to and then SD will sweetly ask daddy and he says yes.

Number 2~ How is his EX running shit in your home? You told your H of your disapproval and he still went against you his wife and agreed to the 4 year old coming over anyway??? Yeah that's a hell NO for me. The problem here is your H who is not respecting and dismissing you. You need to address your H and lay down the law and tell him the changes that you expect to see. He gets away with this because you have allowed it. Find your voice and let him know this is not acceptable. This is YOUR house also and if you don’t shut this down it will persist and things will only get worse.

Just because you are a Christian it doesn't mean people get carte blanche to flout boundaries willy nilly.  Jesus didn't like people messing with his, huh?  he threw the moneylenders out of the temple.    I bet he would hold your purse while you laid down the law on this issue to DH, SD and BM.  

I'm going to problably be in the minority here, but I think that as long as the kid is behaving and cleaned up after, it shouldn't be an issue.  I can see if he was coming over, throwing temper tantrums, breaking things, pulling the cat around by it's tail.   And since you aren't asked to do anything, and you aren't even home, I say this is your DH and SD's home too.   They should be able to invite people over.  You don't get to be the sole dictator.  

I think the issue is that both your DH and SD have poor boundaries with toxic BM.   Let them get run over by her if they want.  You need to pick your battles, and decide where your own boundaries should be, in other words what is worth getting worked up over and what is not.

Your husband is allowing BM to dictate what happens in your home and neither he nor SD have any respect for your boundaries. Your DH obviously has no problem playing step daddy and chauffeur to the boy.

If this boy being in your son's room is unacceptable, get a lock for the door and YOU keep the key. No one else. 

However, your DH is your biggest obstacle.

Biology wins every time. No matter the reason BM lost custody of SD, SD still has that primal urge to please BM by watching her half brother. Plus if it's the only time she sees him, it's the only time to build a sibling bond. 

Is the situation right? No. Is SD's response to BM understandable? Yes

My BS9 is usually at his father's house while the little boy is over and plays in my sons room after I've told SD to not let him in there. 

This would bother me. I'd get a locking doorknob and lock the room when BS is not there. While BS should be expected to keep his room neat, he should not be expected to hide the items the 4 year old is too young for each time he visits his dad.

Yes, BM is taking advantage of SD, DH, and DH's parents. BM is also most likely denying her divorcing husband custody time by sending the child to SD. Once her divorce is final, hopefully, the newest ex has custody during her work time.

and it is not fair to expect her to negotiate with an adult.  I am hoping she gets a job, and that puts an end to this.   I understand DHs fear of his SD being in a tough neighborhood.  

I feel for your SD. She clearly loves her brother, and she has a crap mother who can't facilitate that relationship. My guess is that if she tells BM no, BM will then keep little brother away and crush SD as punishment.

SD is stuck, and she will be the victim of any fallout. Can SD go to BM's house, or can a maternal family member facilitate the babysitting? Or, if new XH isn't a total arse, would he be willing to facilitate SD having a relationship with his son/her brother? If someone else can facilitate, then at least it isn't happening in your home.

Would you be willing to accept one Saturday a month? That would allow SD to work, and keep a relationship with her brother, and limit your DH's excess spending. You could keep it scheduled to when your son is not home, either.

You're doing a good thing, and you're not wrong for not liking it. I'd likely allow it to continue as-is for the sake of my SKs, but I would put up boundaries with my DH about money spent, have very clear house rules with SD, put in some cameras so I know what's going on in my house, and put a lock on any door I wouldn't want folks going into. I'd even likely buy some toys for the boy, but I have the income to do that.

I don't know that I could hate BM enough to hurt my SKs or another kid PROVIDED that all the kids are generally respectful and appreciative, the SKs want to do it, and BM isn't using the child to stir up additional drama. I admittedly have a soft spot for kids, and really, I read this as an actual "for the kids" problem. Sure, BM is using SD, but SD is enjoys it and it's good she has a good relationship with her brother.

I don't have a good solution. There just isn't one. I can see all sides of this, so I'm not sure there is one good or bad answer. I do think tightening up on boundaries is important and making sure DH isn't feeling responsible for feeding/clothing/entertaining BM's son and SD knows that this is a job, you're the boss, and what you say goes. Past that...it's a toss up for me on what is the right path forward.

Put a lock on DS9's door.

If you do not want this kid over your house,  why is he there.  I can understand how you feel. The EX is controlling your home.   Your main problem is your DH.  
 

He is allowing his EX to control what's happing.  He is putting his EX and her wishes, before you.  The adult who house it is.  If SD wants to babysit , then she should go somewhere to do said babysitter.  Not your house.  Time to take charge, and see where you really stand.  Are you more important, or is the EX more important?

Then it  will be time for your future plans in life.

Tell me, what would happan if the boy had an accident while at your house? Would your insurance cover any medical or legal costs?

and upon rare occasion, my DD14 will babysit my DD5 at his house.  I did get his permission.  He knows that DD14 wants to earn money and that I pay her to babysit.  I could easily pay some other teen in the neighborhood to watch my DD5.  She could watch her at my house but it's his custody time that this has occured.  He also doesn't have a lot of money so he is unable to pay DD14 for chores or to get her an allowance.  IF she has her own spending money, then she is not likely to ask him for money for things and it's only been for a few hours here and there.  I always ask him and her and they always know they can say no and that I have tons of other options but if I have to pay someone , then DD likes to get first dibs at the money.  I wouldn't feel comfortable if 1)I had asked him and 2) no one was getting paid.  That wouldn't be right.  DD14 also will babysit some of her stepmom's coworkers children at their house to earn some money every once and awhile too.