The starlink works well on WiFi with its own router.
Got that.
When I set it to bypass mode and connect Ethernet I get much slower speeds when connecting directly to a laptop, and no connection at all when connected to the AirPort Extreme
Sorry, I am not understanding what you are hoping to accomplish by using "bypass mode"......(whatever that is)......when it is clear that this setting is either not the correct setting to use.....or.....if it is, it is not configured correctly.
If the Starlink device works well as a router, it will be a very simple matter to connect the AirPort Extreme to the Starlink router using an Ethernet cable and set up the AirPort Extreme to create a wireless network. AirPort Utility will automatically configure the AirPort Extreme in the correct Bridge Mode setting.......(Apple calls this "Off (Bridge Mode)" just to be different.
Actually, if the Starlink router is working well and providing a Wi-Fi signal for devices, I have to wonder why you want to use an AirPort Extreme, but that may be another discussion.
If you want to use the AirPort Extreme, you can connect to the Wi-Fi signal provided by the AirPort simply by selecting the name of the Wi-Fi network that you want to use on the device that you are using at the time.
Yes, you will need to reset the AirPort Extreme back to default settings, but the very simple setup will only take a few minutes at most. If you need help with this, ideally you will have a Mac to use for the setup, but an iPhone will work.