If I was in your position I would sit back and wait until the Strata Committee escalates this further. In the meantime arm yourself with as much data as possible to support your case.
If you provide a statement at mediation that outlines that: the safety net is not permanent and that it will be removed when your children have reached an age when such a device will not be necessary; and that you will guarantee that you will repair any damage to the common property should it be proven that damage has occurred, then this may assist you in rebutting some of the arguments that have been raised against you.
Incidentally, do you know which architect designed your building? If you do then perhaps you could contact them to ascertain whether they have an issue with you installing this temporary child safety netting. If they have no objection then this may add weight to your case.