10.5 month old wakes up at 2 and 5 am

Looking for advice on how to restart sleep training.

At 4 months when my SO was on paternity leave we did pick-up/put-down and that got him to only 1-2 night feedings. At 6-7 we were able to cut the night feeds and baby was waking up at 6:30-7:30. So for about 1 blissful month he slept through the night.

Then in October he started waking up at 5 am. At first, I was able to rock him back to sleep. But after daylight savings, it became impossible and I broke down and let him sleep on my chest on the couch. I know that was a mistake but my SO works 12 hrs shifts and comes home late, 6 days a week, so he doesn't help at night. And I was getting desperate since I also work FT and basically take care of all the housework and daycare pickup etc. I needed sleep.

Over the past few weeks he's had a bad cold/cough and woke up congested between 1 and 3 am. Would not go back to sleep on his own and needed to be rocked. And still wakes up at 5 am.

This week he is over the cold but he wakes up at 2-3 am, takes 1-2 hours to put down and still wakes at 5. He takes 2 naps at daycare. They used to be about 1 hour each. Now it is more like 30-45 mins.

For context, we live in a small apartment so he doesn't have his own room. His dad snores and coughs loudly. And we can't move, the city is in a rent crisis. I could temporarily move his crib to the living room. But that's not going to work long term as my SO uses it when he comes home from work.

How do I break this cycle of waking up super early? What is the best sleep training method at this age?