For parts that DONT have flat surfaces like you have explained, i would use the oven cleaner, then all my work would be done with my dremel or other rotary tools with polishing attachments and different compounds.
In my experiences from metal polishing here in new zealand and australia i havent come across the tumbling method, so im unable to give advice there sorry.
As for the easiest way to polish it would have to be with commercial tools, either a bench grinder or barrel/straight grinder with polishing mops/wheels. Different compounds and applications will essentially make your work alot easier and an improved finish.

If you dont have the tools, and there is no budget, i would suggest taking it to a polishing company getting them to do it, or you could even look at the option of electroplating/chroming!