Run a pair of Ubiquiti Litebeam. If a WiFi extender is working then a proper p2p radio would work better.

Ddwrt can do this with switching gateways via a script that pings thru each ISP. I do it this way with the other isp in a house a block away connected via Ubiquiti litebeams.

Replace them with some cheaper AP Like a tplink.

I would recommend a semi-P2P solution.

Buy an older Ubiquiti Litebeam M5 or M2.4. Buy whichever signal you want to from your main router. The M5 would be Great if your laptop/phone has good 5GHz signal. Set up the LBE outside the shed where you plan to mount it. Give it an static IP based on your Network IP outside the DHCP range (e.g. the main router is 192.168.1.1 and dhcp gives 192.168.1.100-199, so give your LBE the IP 192.168.1.11). Access it again using the new IP, go to Wireless settings, disable AirMax, and connect it to your main router’s WiFi. When its connected you are good to connect it to an AP (below).

Buy any router you can use as an Access Point. A cheap TP-link AX10 or any spare router your have. You’ll need to set it to Access Point mode or disable DHCP at least. Then give it a static IP based on your Network IP outside the DHCP range (e.g. the main router is 192.168.1.1 and dhcp gives 192.168.1.100-199, so give your AP the IP 192.168.1.2).

Connect the LBE to the new “router” (now an Access Point) via lan-to-LAN ports. Careful not to use the LBE’s PoE port. Voila!