Run an external script from F5

No - there is no simple solution.

The simple solution is that the LTM would expect both pool members to be active and ready to receive requests, even if one of the pair does not get selected (via priority group load-balancing, for example). Then when the primary pool member fails, requests are directed to the next available server in the pool which is already running the required service.

For a more complex solution, you could use iCall, and use that to trigger a bash script that connects to the pool member to make it active:

What is iCall?