Within minutes of posting this, @Jpfamily was able to help me get the answer.
The appropriate curl command is:
curl -H "Bond-Token: (token)" -i http://192.168.x.x/v2/devices/(deviceid)/actions/StartDimmer -X PUT -d "{}"
I wanted to post this so that anyone else trying to find this answer later will be able to. If you have a fan with a single button to turn the light on and off, as well as to dim the brightness, you can send this curl command from your Home Assistant terminal, and it will start to cycle the brightness up and down.
Before sending the curl command, replace (token) with your token from your Bond app. Set the IP address to the IP of your Bond bridge, and replace (deviceid) with the device ID that your fan has been assigned.
You can find out your device ID by going to the Bond Local API documentation and following the “Getting Started” steps. It will teach you how to send your first few curl commands including how to find out your device IDs. Also, I highly recommend using your router settings to set a static IP for your Bond Bridge before trying to do this integration.
Then you should be able to use a service in Home Assistant to send that command, and you can set a certain interval with an automation until it sends a Stop command (that’s what I plan to do).
This won’t be a perfect solution to get brightness to exactly where I want, but at least I’ll be able to get the brightness to change (somewhat unpredictably) via voice commands by integrating Home Assistant with Alexa. Slightly unpredictable voice control is still better than nothing!