Is it permissible to register a portion of the house's ownership to the wife instead of her deferred dowry and its appurtenances, knowing that housing is her right upon the husband?

If the wife agrees to have half the house or a portion of it registered in her name as compensation for her deferred Mahr (dowry) and other entitlements, there is no objection to that. The marital home is not the wife's property; rather, she has the right to reside in it. The husband is not obligated to build her a house for ownership unless he voluntarily chooses to do so. If he does transfer ownership of a part of the house to her, this does not nullify her obligatory right to reside in it.