After some testing, this is how I was able to get the 1011 Captain II to start with the engines running in P3Dv5 first time, every time:

1. Load the L-1011 into P3Dv5
2. Select Add-ons > CS 1011 Captain II > Make sure Panel State: Never Load Default is visible (click on it if it says Panel State: Always Load Default for it to change to Panel State: Never Load Default (I didn't see it change, but the next time I navigated to it, it had changed).
3. Exit P3Dv5
4. Restart P3Dv5, and at the Scenario Screen (make sure the Scenario Screen shows when you start P3Dv5, by selecting (checkmarking) Show on Startup on the Scenario Screen) select LOAD
5. Select 1011 Captain Cleared for takeoff (KVPS)
6. After it loads, the engines should be running. From there, you can select a different airport, change the weather, and the date/time, and still have the engines running after making those changes.

Note. Once you have selected Panel State: Never Load Default, every time you start P3Dv5, simply load the 1011 Captain cleared for takeoff (KVPS) from the Scenario Screen, and you're good go with #6.