Pushing it to my max on a 7% 2kft climb. I'm cardio limited, but legs never hurt. How do I push it harder? I have a climb by my house that is about 2,000 ft over 5mi (610 metres in 8 km or 7.6% grade).

I ride it a least once a week and I've got a ton faster (at 35min right now). Recently my legs stopped burning on it. I'm pushing it as hard as I can, and I think I'm in the 80-90% max HR when I do it. Anything more and my body quickly forces me to pull back.

My cadence is about 80 on it, and if I shift up, I seem to go slower, and if I shift down, my cadence is too high and again I go slower.

I do a lot of rides around 7-8k ft (2133-2438m) over 50-60mi (80-96 km) so this isn't a hard climb, I'm just trying to do it as fast as I can - You know, beat my coworkers.

Should I be able to pull more from my muscles on this climb? If so, how should I change my training?
This type of effort very much comes down to W/Kg at FTP (Functional Threshold Power).  You can get lighter, or you can work to raise FTP.

There is no magic formula to improve FTP, it requires (beyond a basic level of fitness) either a coach, or some well thought out self coaching.  A power meter and knowledge of how to use it is an invaluable tool, but where this is not possible, a HRM (Heart Rate Monitor) and RPE (Rate of Perceived Exertion) can work as an alternative.

A very popular training session is to ride 2x20mins at FTP 1-2x per week.

Self coaching can be tricky, but rewarding.  Pay special attention to making sure your easy rides are easy so that your hard rides can be hard.