For years we have estimated deer weight by measuring behind the front legs of the dressed animal, and looking them up on the girth/ weight chart from Penn State.

Since we already hang them to skin them, makes sense to weigh them either instead or in addition to measurement.
Anyone have a cheap reliable suggestion for a scale (prefer Amazon or other easy source)?

Not sure if you have a Tractor Supply around you or not but they have a hanging scale that goes up to 330 lbs for under $20.00.  If you don't have one around you then maybe there is a local farm feed/co op store close by.  If you're in the City and don't have on of those or a sporting goods store that would have one then look on Amazon.  
Since you're not looking to use the scale for anything other than weighing your game animals it doesn't have to be a state certified scale.  If it's off a few pounds who cares.  I have one I bought at Cal Ranch(similar to Tractor Supply) years ago and it works just fine.  Honestly I haven't used it for a very long time because it doesn't really matter to me anymore.  Once i've shot the animal I have to tag it so it weighs what it weighs and that's it.