This is a common concern. Mixing quite divergent goals into one training plan is challenging to say the least. Steve likes to say: “No one will ever climb 5.13 at 8000 meters”. The training demands of these two goals are so different and so high as to be mutually exclusive.

If your climbing goals and running goals are low enough (and by that I don’t mean absolutely low, I mean low relative to your current abilities) then training for both will not be out of the question.

The aerobic base provided by your running training will definitely enhance your alpine performance on longer climbs with longer approaches. Adding in rock climbing and rock climbing training is the trick.

If you plan a hard climbing session it will be best to do it before longer aerobic work so the quality of the climbing can remain high.

Only experimentation on your self will help you determine the mix that can work for you.
All the alpinists I have worked with do a lot of running in the mountains. In general though they mostly climb to maintain climbing performance not necessarily improve climbing performance. If that makes sense.

It’s just hard to advance the ball in both realms at once and the higher your level the harder it becomes.

Scott