Can you set up a non-Apple WiFi range extender to work in bridge mode with an Airport Extreme?

Joowin extender never appears in AirPort Utility (or does that only show Apple devices?)

Apple is unique.. one off, nothing else invited to the walled garden.. you should know that!!

I can set the Joowin range extender to work in AP mode, but then we can't roam seamlessly around the house, which is the main reason I bought this extender.

You are confusing things.

Wireless extend on older devices like apple airports is not like mesh and does not include IEEE 802.11k v or r which are required for roaming.

Mostly roaming requires several identical brand AP of most recent design (with controller in some cases) or proper mesh and still does not solve handoff and roaming on some devices as it is still a client issue above all else.

AP mode is correct and only one that will work.

I have not heard of the Joowin product.. sounds very Chinese??

You will do a lot better buying a full mesh pack.. e.g. cheapie like Deco M5 or better grade Netgear Orbi. Scrap the Extreme unless you need it as a router.. but both Deco and Orbi will work in that role. And either can use ethernet or wireless backhaul. The manufacturers put a lot of effort into fine tuning roaming which still is not 100% and is unlikely to make it seamless until a couple of further generations of products. Client and Server side.