Can I split the syslog to different rsyslog destinations depending on the partition (name)?

Nope... You'd need to implement something yourself to accomplish this. Probably the best (i.e. most secure/easiest) way of doing this is to have the bigip forward all the syslogs to something like splunk, and give people access to that. (In that way there's also only one place for people to look when you're using a cluster rather than stand-alone device).

H