I would never want anyone to make a decision on this based on my opinion, it's such an important thing that ever one should (I think) read the evidence and then make managed risks based on their own view of weighing up the risks.
There's never going to be zero risk, so it's about juggling and assessing individual factors.
a baby that literally won't/can't stay on his back
... probably has reflux, so GP appointments and prescriptions would be needed first and foremost.
I'd raise one end of the cot to ease this and use a dummy - it helps with reflux.
Then, very personally to me, I'd use the bouncy chair in the short term if sleep on the back is absolutely impossible. The raised head and the ability to bounce helps with sleep.
But I would do this whilst bearing in mind the nature of the risks involved. The main risk of bouncy chair is leaning forward to tip over - a risk lower when baby is very young so immobile.
There is also the risk of suffocation when the head bends forward. So I'd make very sure that the gradient of the chair wasn't too high.
I'd be balancing both of these again at the fact that the single greatest risk of SIDS comes from side/back sleeping. So while the above both carry risk, it's a lower risk than poor sleeping position.
I'd also ensure I did everything that helps lower SIDS risk. Namely:
- dummy use
- room share
- correct temprature and clothing
- attentive to all baby's needs
I want to reiterate though, I wouldn't want anyone to make a decision based on my (or anyone else's) chooses. Risk management is very personal.
Each person needs to make a decision on which they can justify to themselve if the worst hapiens and their baby dies, that they did their best to make the right decision.