It has been a while sense I looked at a Trace Manual, but I believe their default setting would be charge only when plugged in, it should leave the AC in bypass mode and charge the battery. This function can be slightly different on different manufactures, models.
For the Tripp-Lite SMB used it was pretty much like REF said, I know my Magnum Inverter has a distinct charge on button.
In any event with any of these inverters you could be suffering from two things when running just the charger, the Starcool is a heavy use AC item, if there are any other things plugged into the AC at the same time they will also add to the consumed AC power. If this goes up high enough the charging power is automatically dropped down so that it can support the AC Loads. This means it is possible your charge input would be lower than it would be normally (Say if Starcool wasn't running).
The second problem when running Starcool is that it also has a pretty high DC current draw (especially when blower is on high). This high DC load along with any other DC items you might be using would divert a good deal of the DC power from ever actually going into the battery. So it could appear that the battery was not charging.
In any event, if the charger was not working , it would be a pretty serious drain on the battery, and I expect you would find that out before getting back to this post.
hope this helps
2004 E350 EB V10 E/PH 2WD