Rerun- if you have joined the advanced class, you might meet Mimi there. She has two little papillons (Oliver and Samson). She has a training pouch (round one) like you are talking about; it is a pink and black faux fur. I made that pouch for her. I made it by buying a normal one, ripping it apart, and using it as a pattern. I can tell you that of the three that I have ripped apart, those standard pouches (like the ones your trainer sells) are definitely washable. They're just nylon and lining material. I would tumble dry low, though.
RATCHX3 High Jinks vom Neuanfang - Gotchya Day: 1/23/2013 DN, SG, SR, CA
Inspire Spark to Ignition! (Ignite) Gotchya Day: 10/8/2016
C-ATCH Pimg - RATO, UJJ (x2), DJX, HIT, CA, CGC, High in Games, Reserve High in Standard- CPE Nationals 2014