Is anyone aware if this is possible?
I don't have a great answer. Some thoughts:
1. A separate (duplicate) worksheet for each region.
2. Moving Region from Page to the Rows so you get a Row Header for each Region. You can optionally hide the header. Use the indpendent scale option you mentioned. Then increase the cell height (click on a white area in the chart and press CTL + Up Arrow) until it is large enough to force a page break on printing.
Obviously these both have some fairly significant drawbacks.
One other thought:
3. If you are using Tableau Server, you could use TABCMD to get a pdf or image of each separate region.
Maybe others will have some thoughts too.