'suggest that once' is correct and usable in written English.
You can use this phrase to introduce a suggestion, idea, or course of action. For example: I suggest that once Spring arrives, we should plan a picnic in the park.
But it is wrong to suggest that once you've decided you're moving away from stenography, that the choices are easy.
Mr Hagel did suggest that once in Iraq, some troops might cross the border from there into Syria.
Our findings suggest that, once public support is no longer available, the two subsidies have different effects on different types of SMEs.
Some scientists suggest that once the spruce and pines in the boreal forest burn, fire-resistant trees such as aspen will replace them.
All these results suggest that once SV40 enters the cytoplasm, it undergoes an architectural modification that exposes the virion's NLSs for nuclear entry.
Subsequently, these approaches suggest that once implemented these values are kept fixed leading to sub-optimal performance in both speed of adaptation and robustness.
Reports indicating that this decision happened very quickly – with no prior notification of allies – suggest that, once again, Trump was winging it.
These findings suggest that, once initiated, barrage firing proceeds and eventually terminates independently of astrocytic calcium signaling, as suggested by astrocytic depolarization described above.
He was one of the first top editors to suggest that once The Post had obtained a copy of the Pentagon Papers, publication should go forward.
There is good evidence to suggest that once they get into university, pupils from state schools perform better than those from independent schools.
I've spoken to a few people who suggest that once you graduate from university it's hard to maintain close friendships as everyone goes off in different directions.
