As I understand an Access Point (AP) would work to provide WiFi to the shop, with the caveat you'd need to run at least an Ethernet cable to it from a router where service is, (or a network switch. A Power over Ethernet (PoE) injector is provided. It's unclear but I believe you need a power supply to use that, or use a PoE network switch like an Omada Hardware Controller. (I don't have experience with TP-Link's Omada line I'm afraid.
The second option would likely work as I believe you have things set up, as you don't need to run Ethernet from building to building. Also here I haven't had experience using such a bridge.
ASUS has RP (repeaters) and TP-Link has RE (range extenders). These are not business rated but consumer/home rated. These wirelessly extend WiFi signal, vs Access Points, which typically need to be ethernet wired from the source.
You originally mentioned router upgrade option, and a lot of ASUS routers can run as AP, or RE/RP. Most ASUS products also mesh together whereas TP-Link you need to be careful as they have a few mesh systems which aren't compatible with each other...
