Having watched my HR for a couple decades, I think having both power and HR is a very good thing. One works with power zones to increase one's ability to hold power for longer periods and more power for shorter periods. Power doesn't report on your physiology, unless you can call no longer being able to turn the pedals a report. HR reports your physiologic zones. If you keep holding power as your HR rises and rises, pretty soon you can't hold the power. Rising HR is the advance notice of that. Pushing power while your HR has risen out of the zone in which you're trying to perform is counterproductive. You're not stressing what you're trying to stress. Note that is not true of the athlete you're trying to emulate.
I think 2 ea. 24" box fans would be your immediate next step, very inexpensive and effective. It's possible they won't make any difference, but you want to know that before trying anything else. I doubt you're dehydrating during these short workouts, but you could be overheating. Bare torso, right?
Next thing would be to forget about this time-crunched cyclist type of thing and do an endurance block. For the next couple of weeks, put in as much time in zone 2 as you can recover from, preferably 2 or so hour rides outside. You could simply be lacking in endurance. A good goal for decent performance would be ~400 hours/year. Ed Burke said that trying to do high end work without an adequate aerobic base was not productive. IME there's something to that. I start training for the next summer in October and don't start doing hard intervals until February.
Another useful thing would be to get a premium TrainingPeaks account and log all your workouts there. It takes some time to figure out how to interpret the Performance Manager Chart (PMC). I've found being able to chart my body's readiness to train has made a big difference. No canned workout schedule or online training plan is really usable without being able to interpret its effect on your body in real time.
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Results matter