Hi there and welcome to BW forums
I'd tend to agree that he's probably OT, those short naps and a day ~13h is probably more than he can comfortably handle. If he's waking a lot in the evening, ie during the first few hours of his night sleep, that's a classic indicator of OT, as is early waking in the morning. I understand why you want to keep him up for your DH but can I suggest that you get him to bed a bit earlier for a few nights to try and help him catch up on sleep, and then once he's sleeping better, start shifting his whole day later, so he wakes up later and then can stay up to see Daddy?
I have to say, that white noise app sounds great
If you could post what actually happened on a recent day, rather than an 'average' routine, it may be easier for us to suggest timings for you to try. Without this information, I'd suggest aiming for him to be up for 3h before his first nap, so you'd start winding him down a bit earlier, however long it takes, and aim for him to be asleep at the 3h mark. Let me know what happens... If he does a short nap, try to resettle him as you did the other day or however he needs you to - that seemed to work well. If he can't resettle at all, get him up but try to make the second A time a bit shorter, or he'll likely become more OT as the day goes on. Since he settles independently, I'd even use the pram to get a longer second nap out of him if the first one ends up being short - just whilst you're trying to help him to catch up on OT, as a nice long nap, however he gets it, can help to 'reset' him and stop that OT building up. Be aware that moving naps are rarely quite as restorative though as non-moving naps, but if it's a choice between a second short cot nap or a long moving nap, I'd go for the latter at this point, to try and break that OT cycle.
How long is his 3rd CN? In order to bring BT earlier and thus lengthen his night, you could try making it a bit shorter, especially as you rightly say that he should be moving towards dropping it, I'd say by around 9mo at the very latest.
Does that sound like a plan for now? Please don't feel awful, I know exactly what you mean but it sounds like you've done really well in giving him the tools he needs to sleep, it's just hard to break out of these OT cycles and when you're sleep deprived yourself it's often useful to have a fresh set of eyes, so I'm glad you decided to make your first post on here after 'lingering' for so long