A mesh system isn’t, strictly speaking, needed, but turning that extender into a first-class participant in the network is. Swapping the extender for anything that can broadcast an ssid (Access point or any home router) and extending the wired network to it (cat5/6 across the attic/crawlspace ftw) would also resolve the problem, albeit exchanging the cost savings of not buying a full mesh system for labor installing the new wire.
It’s rare that the business would give them the slack to be able to, usually there’s plenty of intent