Can I use the F5 LTM v11.6.1 to act as a proxy for ext router's Syslog?

Yes it can. You did not describe your network so I'm going to assume that the router can reach the Big-IP and that the Big-IP can reach the internal syslog server. If so, then the answer to your question is yes. You can create a Virtual Server that listens on say port 514, and then create a pool with a single member that listens on the same port (the syslog server). Then use the Pool as default Pool for the Virtual Server and that's it. Make sure your Virtual Server has a TCP or UDP profile and no HTTP profile as this isn't HTTP traffic.