Just following up, it looks like adding the line
AllowEncodedSlashes On fixed the issue I was having with the escaping of slashes you mentioned.
Code:
LoadModule proxy_module modules/mod_proxy.so
LoadModule proxy_http_module modules/mod_proxy_http.so
AllowEncodedSlashes On
RewriteEngine on
RewriteRule ^/calibre/(.*) http://127.0.0.1:8088/calibre/$1 [proxy]
RedirectMatch 301 ^/calibre$ "/calibre/"
I am running Apache/2.4.18 (Ubuntu) if that accounts for any differences in our results where you didn't have to add it while I did.
Thanks again for your help!