I have heard many babies sleeping only 20 minutes at such young age. That is one sleep cycle. Do you have swing? It's not ideal to associate sleeping with swing/motion but if he falls asleep (wherever you placed him) and wakes after 20 mins, can you then pop him into swing to see if he continues to sleep? I know you dont believe at such early age habits can develop and as long as he gets some decent sleep, it is the most important right now.
With my bub before I taught him how to self settle (something which is still ongoing) from 3 months, he would sometimes sleep one cycle (20-40 mins) in his cot then cry. I always had be around to prepare for his crying and to give him a bottle to resettle him and then pop him on the swing and he could go on to sleep for a much longer time. So it was cot-bottle-swing. It didnt always work but at least it did sometimes.
After which I got sick of always being on my toes (waiting at his cot for him to stir and not being able to do much in the house) so I taught him to self settle.
Maybe you have tried this but have you put him in a stroller cover a snooze cover/large cloth and wheel him around?