At least a couple of people have been experiencing difficulties with ATS and
transparency. Fixes for TS-804 (requires libcap) and TS-816 (attributes on
other ports don't work) are now committed if you want to give them a try.
I have tested on my setup with full transparency and NAT based outbound
transparency simultaneously successfully.
One note is that if NAT is done on packets before they reach ATS that port
should not be inbound transparent. In this case if you are OK with having
connections to origin servers use the ATS address as the source address then
you should not enable transparency at all. If you want ATS to use the client's
address then you should use outbound transparency ('<'). Also note that you do
not need an external router to use the inbound NAT case, you can do that in
iptables using the REDIRECT target.