I'm happy to try to answer - as I said the other day I can read the desperation in your posts and am genuinely trying to offer some helpful thoughts from your son's perspective.
Essentially, rather than prioritise and focus on me, or even on our mother / child relationship - every time I saw her, or spoke with her, became a chance for her to focus on her, how hard her life was, how awful life had been and how unfair the way her separation from Dad panned out was. In adulthood I can understand why she left the home and why she acted like that in those early years, but as a 13 year old I needed my mum to listen to me, accept that my home and family hadn't completely changed like hers had, and give me the space and time to just 'be'.
What I would have said to my Mum had she been able to listen, remember at 13, your child is just at the cusp of starting to pull away from their parents into their young adult-hood - give them the space to do that - even if that means you see them for just a few minutes a week - it would likely look similar if you were still at home , give them the confidence that you are there if they need and give them the respect that understands they are neither to blame for, or responsible for, any of the actions or decisions that you and their Dad have taken.
I'm sorry you've found yourself in this position, particular after what sounds like an awful marriage - beyond anything take ownership of leaving, and the impact (both good and bad) that's had for your son - giving him that respect will hopefully be the basis for a good relationship in the future, even if he never chooses to stay the night in your home.
I know you and his Dad planned 50/50 custody - but was he involved in that plan? If not, think of why you didn't do that - he's old enough to have his views about where and who he lives with listened to - perhaps you'd not be where you are in any of you had asked what he wanted. As many have told you here, you can't force him to sleep in your house, you can't force him to want to visit - ask him what he wants and go from there.
If it can give you some hope, since the third month after she left Dad, I've never stayed the night at my Mum's - but we are close now, have been since my early 20's - it's different to what many may expect a mother - daughter relationship to look like, but it's a rewarding and meaningful relationship in my life.