Yes climbing definitely messes up your heart rate and I can imagine this is getting in the way of your zone 2 training, I've never thought about that fact and makes me appreciate living in a flat country more 😆
Indeed, power meters are very expensive.

Without a lab test for heart rates, it's all guessing. My 70% (top of zone2) is 123 BPM and that's definitely too slow, but it's just a base value and your aerobic threshold gets higher when you're fit.

The aerobic threshold (also known as LT1 or VT1) can be significantly increased through training. In untrained individuals, this threshold is often around 60–65% of maximum heart rate, whereas in well-trained endurance athletes it can rise to as much as 80–85% of maximum heart rate.

That's why I go based on feeling and the facts I can still talk without running out of breath and breath through my nose.

Pretty interesting, all the science behind it. It's something I've never thought about when I started cycling.