I don't think anyone can realistically answer this with any certainty. There is always going to be a risk and it is for you to decide how you manage that risk.
The sleepyhead is not the "firm, flat mattress" that SIDS recommendations advise for baby sleep. I also doubt it would adhere to the "Clear Cot" recommendation of no pillows.
However that does not make it inherently dangerous, just an increased risk. Take, for example, the fact that cosleeping carries an increased risk of SIDS death. However cosleeping is not considered dangerous, not even within official SIDS guidelines. It is about manage the risk as much as possible, not eliminating the risks to zero.
There will be a risk attached to using a sleepyhead, and an increased risk compared to flat mattress. There is a risk attached to everything to varying degrees. Some people are risk averse and will only consider ways to minimise risk. Other people will balance the benefits against the risks and make informed decision to manage the risk.
So let's not pretend there is no risk of suffocation and smothering. There will be similar risks as associated with using pillows I would imagine*. But that does not mean that it could not be a managed risk to an individual or that the sleepyhead is inherently dangerous to use.
- I am using my common sense in making that statement, rather than having it quantified by research