You’re wise to treat these two sports as distinct in their demands and training approach. While there is come crossover when it comes to the core strength area much of the training will be different enough to even have a negative effect on the other sport unless planned well.
It sounds like you have determined that you have some foundational strength work to do that will help your running. Those exercises could easily be incorporated at the end of climbing gym session when the climbing training is a priority. Hill sprints are the most effect way to build leg power for mountain runners, if you have a steep >20% grade hill with good footing.
Plan your hard climbing sessions, indoors or out when you are well rested. It is fine to do a longer run after a bouldering session but doing a long run before you go bouldering you will not have the power you need to get much out of the bouldering.
Remember: Strength, Power and Speed always before endurance.
I’m going to ask Dave our climbing coach to chime in with some specific ideas about who to improve your climbing weaknesses.
We do not have any stock plans for this combo of running and climbing due the immense variability of the user group but we’d be happy to discuss putting a custom plan together for you or a coached program.