I'm looking for a standard, or sound industry practice, for the equivalent of a Key Check Value, applied to the private key of a public key algorithm, like ECDSA.
For DES or 3DES keys, practice (as worded e.g. in EMV Card Personalization Specification) is:
6.1.5 Key Check Value
Purpose: The data is used to prove that a card/processor has access to a specific DES key value.
Format: Binary, 3 bytes
Contents: The three leftmost bytes of the result of encrypting eight bytes of zeros by the DES key concerned.
Notice that the above definition is slightly wrong: the KCV as defined does not prove anything beyond knowledge of the KCV. The actual purpose (and the one I'm interested about) is to guard against a wrong value of the key, either accidental, or deliberate from one participant in a scheme where the key is rebuilt from shared secrets (for simplicity, assume that's by XOR of the components, and that at least one participant does not cheat).