Can I transfer my RaceChrono Pro purchase from Android to iOS? Or vice versa?

Unfortunately this is not possible. The purchases are transferable between devices with the same Play Store account or App Store account, but not between accounts.

There are technical reasons and contract reasons:

1) License systems in iTunes App Store and Play Store are completely separate, and not in control of the app developer who doesn’t even see who has purchased the app. He sees only transactions with vague location, used payment method and amount.

2) Both stores require in their contracts, that not a single app is sold without them getting their fee. So even if there was a way to transfer licenses, it would not be allowed.