The MENU­ITEM­INFO structure lets you specify a bitmap to appear next to the menu item. Is there a way to do this from a menu resource template?

No.

If you look at the format of menu templates, you'll see that there is nowhere to specify a bitmap.

Which kind of makes sense, because it is the responsibility of the application to destroy the bitmap referenced by the hbmpItem member when the menu is destroyed, but if you created the menu from a template, you don't know what that handle is, so you can't destroy it either!