Is it permissible for a wife to send pictures of her purchases, which might lead her sisters to pressure their husbands to buy similar items, knowing that this causes them discomfort?

If your complaint is true, then your wife and her sister are mistaken, as a husband is not obligated to provide beyond his wife's sufficiency, according to his means.

Advise your wife to be gentle, and remind her of the etiquette of Islam in considering the feelings of others, just as the Prophet, peace and blessings be upon him, did when he apologized for not accepting the gift of Sa'b ibn Jaththamah to console his heart. "Religion is sincerity (or advice)," and the wealthy should not display their wealth in a way that breaks the hearts of the poor or burdens them with what they cannot bear.