Aren't your skids in their teens? That is plenty old enough to understand that it's inappropriate to invite their mommy in to play with their new puppy. Also, though, BM should be told to stay out of your house, stay out of your backyard, and not to hang out in your driveway. 

Ugh I feel two ways about this: 

objectively I feel like yes, you're overreacting because all kids are excited and want to share their excitement with anyone and everyone when they get a new anything but especially when it's a new pet. So I'd allow them to have that moment of excitement for them. 
 

but on the flip side, and there always is when there's a intrusive BM involved, I would put the puppy up for a nap whenever BM is around or when the skids want to share him with her. But this is coming from my own personal experience with BM blaming my dogs death (SS left the door open when she was picking him up Christmas Eve and he followed him out into the dark and was severely injured by a car) for SS's "severe depression and nightmares" now over a year later and conveniently just after SS told her he didn't want to go with her for the weekend. Now he's depressed and it's the dead dogs fault. She's a piece of work so I wouldn't share an ounce of joy in my household with her. Tell your DH the puppy is yours, not the households and you will do whatever you have to keep her from inserting herself into this memory. Take the dog for a drive or a walk during drop off or pick up and text to make sure she's gone if you have to. That's what I would do. But again, my past experiences make me overly sensitive. 
 

If BM can't be trusted to not go in your house or manipulate SD into letting her play with the dog, then pick-ups should not be at your house during a time that you aren't home.

Change the pick-ups and drop-offs to weekends, evenings or someplace neutral so SD isn't caught in the middle - mom really wants to see the puppy and she doesn't want to disappoint mom because she fears losing her - but she'll get in trouble with you guys if she allows it. That's a lot to put on a 13 yo.

ETA: We would have never in a million years allowed SS to be home alone when BM was picking him up, hell no. Didn't trust either one of them.

If she only lives 10 minutes away, why are pickups even happening? Why can't the kids just walk between houses?

Thankfully my dog had the smarts to dislike BM greatly.  SO the few times she stopped by and left the driveway (which I put a stop to) my dog would stay about 5 feet away barking. More than lowkey proud of him. lmao.  I'd tell him to hush of course, but i was beaming with pride on the inside.

Some of these things you have to ignore.  Her stopping by is obviously SUPER inapropriate. Taking the dog out of the equation entirely it's still not okay.  Those are the kinds of things your SO need to make known are not okay and then enforce it.  If SD asks if BM can come over to play wiht the puppy, the answer should be no, however, some exceptions.  If BM is picking up the skids and the puppy just so happens to be outside, you can't really say no to her petting the dog without seeming the like unreasonable ones.

I do remember when one of the dogs had pups and Psycho (BM) trying to play with them, how absoltuely angry it made me.  The issue typically isn't a Psycho problem though.  Normally it's an SO not enforcing rules problem. 

I think it's smart that you're setting rules in advance to head it off, just know it falls on your SO to then enforce said rules.  Anything else.  Ignore the wh0re.  THe skids ask to bring the puppy out, tell them to ask their dad, and then he can be the bad guy who says no. (make sure he's aware of the rule. lol)