It's not your job to watch his poorly behaved kid, and it's unfair that he agrees to BM sending him over whenever she pleases. Put your foot down that he's not there unless DH is there to parent him. 

"DH, NO. Your child is unruly, and until you can parent that out of him, I will NOT be responsible for him. If you want him while you work, then you need to put him in daycare. I will not babysit him."

Your DH needs to figure out how to get him to behave in your home. Part of this is age and going back and forth between homes with different rules. At 4, it's hard enough to follow just one set of rules, much less two, ESPECIALLY when the other set of rules rewards poor behavior.

Aside from telling SS he has to be nice and respectful, what does he do to get SS to behave? Take away toys? Time out? Your DH needs a reward-and-consequence system put in place for SS, and he needs to enforce it.

It's time to start putting your foot down with DH. 

No it is not your job to watch his first born.

But depending on finances you may or may not be able to afford a sitter.  And it's not BM's job to cover Dad's slack.  If they have an agreement that SS goes to Dad's on x,y,z days, then their financial agreement likely follows it.  So Dad needs to figure out how to pay for a babysitter, or surrender that time and the associated support that goes with it.  That's the cold hard financial part with no sensitivity to the relationship between DH and his son.

If on the other hand, part of the reason you are able to stay at home with your newborn is because DH is paying all the bills, affording you this time in which you do NOT have to put your newborn into child care (think of how THAT would impact you!), then perhaps telling DH to shove it is not the most appropriate recourse.  A four year old can be a terrorist, but with time and a firm hand you can instill the discipline you want him to have in your home for the rest of the 14 years you will be putting up with each other.  Get them while they're young, and you have a chance.  Try to raise them past the age of 9 and you're tilting at windmills!

As long as DH backs you up that when the little guy breaks a toy, a new toy does not replace it.  Not one from BMs house, not one from your budget.  He goes the rest of the week with fewer toys because he broke one.  Or possibly the rest of the month!  Reward good behaviour, punish bad behaviour and remember - he's just a four year old child.  As a general rule, once they know they are NOT in charge, they are happier.

If you have the money for a babysitter, perhaps it will give you the time you need.  But unless you have him full time, you seem to already have lots of one on one time with your wee one.  I assume you two talked about this before you decided to have a baby and stay home.  And that you may have agreed to take care of the little one while he continued to work.  The reality is turning out to be less than sunshine and rainbows, but your little one is here.  It's not fair to just say you're not doing your part unless the two of you have the money to make it a reality.  If DH is paying all the bills so that you can stay home with your together child, perhaps helping with his eldest child is something you can do to lighten his load? 

I'm not saying it's your obligaion.  And my sympathy for the sucky days that a newborn and a toddler bring.  Many women have wanted to string up their own firstborns in this scenario!

Nope, that child is not your responsibility.  Your husband needs to know how you are feeling.  If you don't tell him he won't be given the chance to fix things.  You'll end up resenting him and SS if you don't already.  You wouldn't watch a neighbor's unruly child.  Your SS should be held to a higher standard than a neighbor.  Tell your husband that he either corrects his child firmly and consistently and backs you up when you do, or you will not watch him.  If he has to give up time that his son would be here, then so be it.  He isn't there to look after him anyway right?  So what does it matter to him if he's with you or BM?  BM most likely is contributing to the problem.  Let her deal with the little monster that she is creating.  

As far as tantrums go, he should be growing out of that stage about now.  My DS threw horrible tantrums when he was 2.  He never threw toys and broke them though.  What worked with him was being ignored.  I would purposely ignore him or better yet send him to his room.  I'd tell him he could come out when he was done with his fit.  And a few minutes later he would come out and tell me he was done with his fit.  Then we would talk about it.  I'd make him apologize and encourage him to use his words next time instead.  He stopped his tantrums at around age 3 I think.  He was doing it for attention.  He stopped because he wasn't getting the attention he wanted.  

If your SS breaks a toy, DO NOT replace it and your husband should not either.  You can't control what goes on in BM's home but you can control what goes on in yours.  He needs to learn that if he breaks something at Dad's house, he doesn't get a new one.  Maybe when all his toys are broken he'll learn.  You may not be his mother but while he is in YOUR home, he will treat you with respect.  You wouldn't tolerate the neighbor's child disrespecting you in your own home, why should you have to tolerate your husband's child disrespecting you?  You shouldn't.  So either he shapes up or his happy little behind can stay somewhere else while DH is working.  Remember he is NOT your responsibility so therefore you are doing your husband a favor by babysitting him for free.  

What does his CO say? What time does he officially have with SS and what CS is he paying to BM?

I think you just have to be honest with your DH. Tell him that his child's bad behavior is making it impossible for you to handle the responsibilities of both SS and your LO, that is has been getting worse, and you don't see how you can continue with this arrangement.  If you don't lay it out clearly to your DH, nothing is going to change, your resentment will grow, and relationships will be damaged. 

At 4, isn't he old enough to go to preschool? My "deal" with DH is that in order for me to stay home with our bio,  I need to also care for the SDs during DH's time (we have 2-2-5-5). That's the only way it works financially.   Lucky for me,  SDs are fairly well behaved, AND when bio was born last year SD8 was in school all day and SD5 was in half day 4K. Having that break from them for a few hours helped when I was so busy with a baby. 

Perhaps not I do not like your child but I do not like your child's behaviour and I am finding it difficult to cope with him. He is disrespectful to me. I am concerned that.....

Can you ask your DH to put him in daycare during his custody days? Of course, that could mean he can no longer afford for you to stay home and you'll have to get a job and start contributing. 

Stat!!

How did it get to this point?

Do you have family near by?

I would give dh a 2day notice of your intentions. Starting today tick-tock.

You and BM have now been put on notice. I HAVE decided that I will no longer babysit your son. WHEN the doorbell rings and you are not here dh, I will not answer it.  On the off chance your x drops off her son while you are here before you leave for work, you better have a babysitter lined because I will be leaving with our son before you leave our house.

Got it, good,