I had decent luck with a really tight spiral on circular areas, which might work for you in this case.
I went deep into it here, there are caveats, and you’ll want to know them:
https://community.glowforge.com/t/super-manual-engraving/15965/?source_topic_id=16086
Inkscape’s spiral tool is nice, and you can actually set an LPI if you are handy with math. Be sure to set divergence to 1, so that the spiral doesn’t go “nautilus” on you. Also, set a very very thin stroke or it’ll look like a blob.