-Ear- is voiced differently in heart, dearth, pear, ear. I don’t see why any of these sounds has a priority claim to be replicated in the pronunciation of “Peart”.
In any event, to approach the question that way is to get things the wrong way around. How does this guy say his name? Right, what would be a good way to right that down?
Heart is the only one of those with the same number of letters and with just one letter substituted. A simple one-for-one trade. But, having said that, you are right that any person or family has the right to choose how to pronounce their name. And the English language is full of such pronunciation inconsistencies.
Well, there is the dialectal word peart which, actually, is pronounced as Neil pronounces it. (Although “pyert” is also given as an alternate pronunciation).