Solved! It was indeed a DNS issue. I have two DNS servers configured, the first one is the Synology box the container is running on, the second one is elsewhere on the network. And for some reason, the container can reach everything on the network except for its own host. (I think I’ve seen that before)
For now I’ve just removed the host from resolv.conf so it always immediately goes to the second server. And things are running as fast as expected now, with the extension connecting in well under a second.
Thanks to both of you for helping me fix this!