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No, you can't put the badge on the toolbar icon. To make a feature request, send it in with Help->Send Feedback or email

I think you might have misunderstood my point about the due soon and overdue counts overriding the display of the inbox count. When you turn on those displays, you get a number in a colored oval next to the project/context/inbox name in the sidebar.

The Inbox count is in gray, and shows that I have 29 actions in my Inbox. Update iPod touch has an orange count, which is the default styling for Due Soon items. AAA test project has a red count, which is the default styling for Overdue items. You only get the most important count, so Overdue overrides Due Soon, and both override the Inbox count. The Inbox item count will always be present provided you have some items in the Inbox and none of the items in the Inbox are Due Soon or Overdue. Every action in every project outside the Inbox can be Overdue :-)

So, to sum up, as long as you don't let anything in your Inbox get too close to its due date/time, you can have an on-screen indication of how many items are in the Inbox.