You are on shaky legal grounds, I think. The Owners Corporation has authority over common property only, and in the case of installing an AC that would likely mainly involve a couple of holes through the walls for the pipes and electric cables. You don’t say if you are an office holder on the strata committee, but only the secretary (or delegated strata manager) has the authority to issue any demand email to the AC installation applicant. Decisions about renovations are made by the strata committee at a properly convened strata committee meeting, and not by individual committee members.
You should have sent that email to the strata manager and strata committee secretary, and request they arrange a strata committee meeting, with the substance of your email as an agenda item. Then the committee could make a decision about sending those demands to the owner asking to install an AC.
Furthermore the renovation is likely to be a “minor” renovation, which has a lower approval standard – your allegation about it requiring a special resolution is likely not correct. If the matter had been dealt with in a strata committee setting, a better solution would likely have resulted.
Your demands, as well as not being lawful, are onerous, and could very well result in the owner ignoring the committee entirely. If they ignored the committee and just installed the AC, then all the committee could do, would be to allege some of the AC installation was illegal under strata law. Assuming the AC was installed by a regular AC installation company, it is likely that it was done competently. The Tribunal, should your committee try to take action, would find in the other owners favour, in all likelihood, and could order damages, if your committee was foolish enough to take action (assuming the AC operated normally, and did not cause any problems, which most AC companies are familiar with anyway).
I have to emphasize that individual strata committee members do NOT have any power to communicate with owners, or to make demands of them, outside properly convened strata committee meetings, although the secretary of the committee may have some rights to do so, but they would be wise to follow the advice of your strata manager, and only do so from decisions made at the committee meetings.