I had this issue with DS1 - he wouldn't sleep longer than 30 mins in his bassinet without waking up. My mother suggested I try placing him on his tummy (as mothers were advised back then to prevent babies from choking on their vomit). I was quite apprehensive the first time I tried it with DS1 (what with all the SIDS info etc)- but I have not looked back since. That day was the start of sound sleeping habits for my DS1, and the start of getting my sanity back.
My second son has been exactly the same- even when swaddled, I believe babies just feel so exposed on their back - its just them and the big wide world - iykwim? They like to feel nuzzled up against something - (if not your chest), to feel secure.

Of course I make sure my baby's mattress is firm and that there is nothing in his bassinet that may obstruct his face. It also makes for a much easier transition from being in the sling. You may not feel comfortable trying it - but I had to share my 'secret'. I would be completely bonkers with 2 under 2 were it not for tummy sleeping!!