oh, the WhatsApp lead gen merry-go-round. i've ridden this ride more times than i care to admit, and yes - it's the Conversions API that saves your bacon.
Basically, when a chat gets tagged as qualified or closed in your CRM, you fire a custom event back to Meta with the hashed phone number as the match key. use whatever event name makes sense - lead_qualified, deal_closed, god_please_work - it's all the same to them.
Set up a Conversions API token in Events Manager, hash that phone number with SHA256 (not SHA1, not MD5 - trust me, I've made that mistake and Meta laughed at me), then send a server event with the event name, timestamp, and user data.
If you're using something like Respond.io, it can handle the CRM side and connect to the API via webhook or Zapier on top. Makes life a hell of a lot easier than manually hacking endpoints together.
Just don't forget to test the damn thing before you pat yourself on the back. i lost a week to silent failures once - never again.