Sounds like their real problem is that they don’t want you to put down floorboards. Presumably your plan is to lay the flooring with proper, top grade insulation underneath (which would be the smart thing to do if you don’t want your neighbours to force you to lift the floorboards and do it all properly).
Replacing carpet with timber – or even just exposing existing floorboards – is considered a minor renovation under the strata Act. The committee can approve it, provided they have that authority from a general meeting (they probably do). The approval may be subject to reasonable conditions but cannot be unreasonably withheld.
So where does that leave you. I would consider seeking mediation at Fair Trading prior to seeking orders from the Tribunal. At least at mediation or NCAT, the committee would have to show any evidence they have that removing the carpet is a health hazard. You could offer a compromise that you will permit specialist removers if the strata scheme pays for it.