Recommended upgrage BIG IP LTM IOS

Don't upgrade for the sake of upgrading. Unless you really enjoy lots of free time right now. Normally, to upgrade, you must have a reason. Do you have a stability or security problem with your current version? Fancy try out a new feature you saw in release notes? If not, there's nothing wrong with sticking to 11.5.4 for another year.

If you insist on upgrading, current conservative LTS pick is v12.1.3. If you prefer not upgrade BigIP to a new major release more than once per 2 years, going with the latest 13.1.0.1 LTS release could be the better option. It's cutting edge with a slightly higher risk on stability. I'd still wait until it's at least a month old before putting it on a production system.