Basic HTTP Stream Profile

The STREAM profile doesn't really work well with URLs. It works on the data in both directions, so basically you're saying replace any http:// references with https:// on the way in (Host headers usually), and again replace any http:// references with https:// on the way out (Location headers and payload object references). So assuming you only want to replace content in one direction, you necessarily need to do STREAM in an iRule. Apply an empty STREAM profile to the VIP and an iRule like this:

when HTTP_REQUEST {
     tell server not to compress response
    HTTP::header remove Accept-Encoding

     disable STREAM for request flow
    STREAM::disable
}
when HTTP_RESPONSE {
     catch and replace redirect headers
    if { [HTTP::header exists Location] } {
        HTTP::header replace Location [string map {"http://" "https://"} [HTTP::header Location]]
    }

     only look at text data
    if { [HTTP::header Content-Type] contains "text" } {

         create a STREAM expression to replace any http:// with https://
        STREAM::expression {@http://@https://@}

         enable STREAM
        STREAM::enable
    }
}

This will only replace http:// with https:// on the way out (to the client). Presumably you have this applied to a port 443 VIP that is offloading SSL.