Is it religiously permissible to conceal a revocable divorce from the family on the pretext of the possibility of reconciliation, given that two menstrual cycles of the waiting period remain, and the husband refuses to inform the wife's family?

There is no blame on the wife for concealing her divorce from her family for any reason, and this may even become obligatory if it involves a benefit. Concealing or announcing the divorce does not affect the validity of the raj'ah (reconciliation). The raj'ah during the iddah (waiting period) is a right of the husband, and its validity does not require the wife's knowledge, her consent, or her family's knowledge, by consensus of the scholars.