When you say school equipment, what exactly do you mean?  Books?  A computer?  Headphones? Other than pens, pencils, notebooks and binders, that's all the "equipment" my skids have.  We do not have two sets of books or chromebooks or headphones, but we do have basic school and office supplies (pens/paper/pencils/highlighters/staplers, etc.) at both homes.  School stuff goes back and forth between the two houses in the kids' backpacks.  We do get a list of school supplies to buy at the beginning of the year, some of which are for the kids and some of which are for the classroom.  BM and DH usually split up the items on the school supply list pretty equally and they each buy some.  It's usually not enough of an expense that either home would ask for reimbursement if they had to buy the whole list. 

My skids aren't developmentally delayed, but they're young (elementary school) and they have more transitions (2/2/5/5).  We do exchanges at the bus stop because except on school holidays, they take place either right before or after school.  The skids have not had a problem getting their school stuff to where it needs to be.  Now, they don't have as many (or as heavy) books as a high school kid would have, but the process is the same.  Frankly, at many schools the books or laptops/chromebooks are supplied, and the school is not going to supply more than one set.  It seems to me that it would be wasteful to have two complete sets of books, and the SS needs to learn some responsibility, whether he's developmentally delayed or not.

Is the BM high conflict, such that she would prevent you from going to pick up a forgotten item at her house?  Has keeping track of things going to and from school been a problem for this SS in the past?

edited to add:  How old is SS?  Depending on what grade he's in and whether he has an IEP, you might be able to have someone at the school do a checkout for him at the end of the day or the end of the week to make sure he's taking home what he needs to.  There were definitely kids for whom this was done, even in high school for IEP kids, when I was in school.

I have two views on this and have done it two different ways - for two different kids in the same house.

My step daughter, we always had one of everything, split the cost against the houses and it came back and forth on Friday afternoons after school. Depending on how big something was or how much she had (musical instruments, text books, laptop, sport shoes, etc), the parent whose week it was to recieve the kids would collect from the other parents house (usually). 

We started the same way with my step son, however he had an inability to take any responsibility for his own belongings and ensuring they were packed when leaving his mum (no matter what his age 8 - 15), and unlike me who would make sure all the stuff was packed for his mum so she didnt have to come back during her time to pick anything up - she did not do the same. So there ended up being so many mornings we would wake up for saturday sport Sans shoes, jerseys, mouth guards, swimming costumes, computer chargers - you name it. So for my step son we employeed a buy double of everything - EXCEPT really expensive items, like his laptop. While this might seem like a waste of money the amount of things that went 'missing' at his BM's or somewhere in between was out of control and we were always on the hook for half.. so this way, no matter what was lost over there, it wasnt on us to pay for it.

I guess it depends on what you mean by equipment. General school supplies should be at both houses. If you're talking textbooks, you can always call the school to see if you can rent a second set or buy/rent a second set online/directly from the manufacturer (many have online codes now so you don't have to buy a physical copy). If he has a school-issued laptop, again ask the school to rent a second one or find out if everything is done online through a learning management system that can be accessed on any laptop/computer.

If you can manage to get all of that, then SS only had to remember assignments going back and forth. With BM and your DH there to assist him, that should be very manageable (because he's going to have to manage all that between home and school anyway). 

If your DH wants to make this happen, he just needs to contact the school and see what they say, but don't be surprised, either, if the school says part of learning for kids is learning to take responsibility for their own things. You've mentioned that your SS has developmental delays, but depending on those delays, being forced to remember to bring things back and forth may actually be designed to help him become more independent.

A back pack.

He can keep all of his school books and supplies in it and carry to school every day.

Keep it simple.