Lindsay, your LO sounds so similar to how mine was! I remember how exhausting it was, and how hard to stay positive when I'd go out and there'd be all these babies who were so much calmer than mine, and if they got tired, they'd just fall asleep in their mother's arms.... But you know, at 16 months now, he is super happy and confident, way more so than those same babies who now cower on their mum's knees when we're out!
I managed to get Lachie to a stage where he would do 3 good 2 hour naps a day, so it is possible! LOs are all different, but here is what worked for us:
We APOPed like anything to get his body into the habit of sleeping, then once that happened we worked on removing the APs. He slept in a hammock which was in our office as I worked from home. I would bounce him in the hammock to sleep, then at the slightest stir would bounce again to resettle. This worked really well as long as I got the timing right.
Double swaddling - we used 2 light muslin swaddles (so it wouldn't matter if they went over his face) the underneath one we'd do an Aussie swaddle and then a normal wrap on top. That kept him pinned in nicely, and the Aussie swaddle give them access to their hands to suck through the wrap.
Reducing stimulation, he was really sensitive to new things and sudden noises, so I kept the house really quiet, no radio or TV. I have a thing about toys, so he only had (and still has) a few toys. He would get really OS when we went out, so I would always wear him when we were out, and for about 1.5 months when he was really bad, we didn't go visiting. If people asked me out, I explained he was going through a sensitive patch and was a lot more comfortable in his own space, so could they come and visit instead. He loved our front pack, so I wore him around the house a lot, that seemed to calm him down when nothing else would. He also would go crazy if he was sitting up somewhere that wasn't snuggling with me or DH, so I just left him on the floor instead. I guess they can't see as much that way, maybe?
BT/nap routine that was the same each time so he knew what was coming
Not clock watching, but just whipping him into bed at the first yawn or eye rub. I actually was putting him to bed too early, because of his OS he'd melt down and I'd assume he was tired. Actually he wasn't, he just couldn't cope with all the stimulation. Keeping things really low key extended his A times which extended his naps too. I don't know what other people think, but I wouldn't worry about games at 4 months. Going for walks in a front pack and being sung to, read to and cuddled is all I tried to do at that age. He had a minimalist play gym that he lay under while I had a shower, but other than that we didn't really do games.
Hope some of those ideas help, hang in there!