These are fantastic questions. First one I’ll answer is “or does it not really matter” the answer is: Yes, it does matter so I’m glad you asked. This is one of those very individualized situations, based on your past with Rainier you do well with loaded hiking/mountaineering and very likely have done a good job training for it thus far. That being said we wouldn’t want to change that training and just train for the run as you would then be less prepared for Rainier and other peaks. However, as you mentioned running is not your strong sport, again potentially due to lack of adequate preparation. This is “general” advice. If you choose to have these two goals with in this time frame I’d begin some running conditioning now. Not add in more training but balance your aerobic work between running and weighted uphill so you have a base to then tweak and build on once you are done with the Rainier Climb or other summits. However, my personal advice if you’d like to avoid the running injuries and do well in the traverse would be to just focus on training for that and take out the mountaineering activities for this season, as we mention training for two different sports/events at once is nearly impossible to do well, especially a big volume like 25-30miles a day for a week that is a significant commitment. Over the next few months you can always reach out to one of the coaches and discuss this or any personalized needs via coaching calls.
I hope this helps!