IANA time zone ID to BCP-47 using ICU4C

Question

Given an IANA time zone ID, such as "America/New_York" or "Europe/Lisbon", how can I obtain the corresponding BCP-47 time zone ID, such as "usnyc" or "ptlis", using ICU4C?

These values are required to generate Unicode BCP-47 Locale IDs with extensions, e.g. 'en-US-u-fw-mon-hc-h23-tz-usnyc".