Thanks for posting this query.
It is very interesting from a coaching/training perspective. You are smart to think first of the basic aerobic capacity needed for this ‘event’. Any event longer than 2 minutes relies primarily on the aerobic system’s ability to produce ATP. The more aerobic ATP that is produced the less you will have to rely upon anaerobically produced ATP that comes with some significant downsides, the best known of which the accumulation of lactate. But there are other issues when one has to rely upon anaerobic ATP production. So, aerobic base conditioning will play an important role in preparing for this speed record. But that role will be primarily in support of the high intensity training you will also be needing to do.
That’s because your event specific work, due overall intensity of the it will done well above the aerobic threshold. So you will need to include event specific training. That is; workotus that look like the actual event or at least parts of it (more on this later).
Training for this kind of thing has a lot of similarities to the training for other high intensity endurance events: Think of the 800 or 1500m track races, the Skimo or Cross Country Ski Sprint events even the running vertical K events. Each of these things, especially the very short ones like the track and ski sprints are only a few minutes long. Yet proper world class training for them is comprised of about 80% low intensity aerobic bace work with the remaining 20% done in event specific training.
This aerobic base of support will allow you to recover from the run/scramble approach which is 5+ minutes of near max effort while climbing. If that approach scramble takes too much out of you then your climbing speed will be affected and certainly the running descent will as well.
Your base training should be done on similar terrain as that approach but at low intensity. Think 3rd flatiron or other long low 5th to 3rd class terrain you can do sustained long workouts on. Steve used to do this workout on WA pass. Hike/scramble the steep 2000 foot Spire Gully to the base of the South Arete of South Early Winter Spire. Then climb up and down that 700 foot 5.5 route multiple times as a climbing specific aerobic workout.
Assume you guys already have a solid aerobic base, I’d do climbing specific base training like this 2-3 days/week with recovery aerobic workouts like easy runs 2-3 days/week while keeping your climbing workouts short and fairly hard to keep your on-sight grade sufficiently higher than that needed. Do this for 2 months. In the second month add in 1x/wk longer Z3 uphill runs (Flagstaff, Greenman Trail) where you pin your HR at the max you can sustain for 30+ minutes.
Then 2 months where you introduce event specific training 1x/week. This will be steep hill hill scrambles (better because they are quadrupedal) or steep hike/runs. These will be done interval style. 2-5 minutes in duration at that same near max intensity.
During this period switch the longer Z3 to specific workout. Again on 3-5th class terrain but no longer at low intensity. Shoot for 30 min/week in Z3 on specific terrain as the approach.
This advice is stated this way because you say, and I agree that the approach is the key to the event. Nail that without putting yourself in a big anaerobic hole and as long as your technical skills and speed are sufficient for the route you should have your best shot.