This may be something of a necropost, and I apologise for that, - but having recently purchased a HomePod I seem to have run into the same problem.
I also use Eavescrop, it's a terrific app. If I recall correctly, the music doesn't have to be _downloaded_ to your device, - however it _does_ have to be added to your Apple Music library (in other words, random tracks that would show up with a "+" sign when viewing them wouldn't be scrobbled).
There are a number of AM clients that have Last.fm scrobbling as part of their feature set (Marvis, Soor, Albums). These scrobble as you play. If you have Eavescrob installed as well, it will capture those separately, but it looks like Last FM is able to filter out duplicates in these cases, and not add things twice.
I'm still not 100% on how well the HomePod scrobbles are captured with Eavescrob though, - it seems a bit patchy, and with a bit of a delay (which is probably the result of syncing, and if that's the case that's fine), but kind of works?