Exchange hybrid (mailboxes in cloud): where to add secondary email address

We are an M365 tenant in Hybrid mode, with only Exchange Online mailboxes, synching mailbox attributes through Entra Connect. We have acquired another firm (not an M365 tenant). We have taken over their DNS and copied their old firm mailboxes into EOL mailboxes on our tenant, which already use our own custom domain as their primary address.

At this point we just want to add their old domain email addresses as a secondary smtp so they can receive mail replies to old threads directly in our tenant.

Initial testing: If we add their old addresses now through Set-ADUser userMailbox -add @{ProxyAddresses='smtp:[email protected]'} and sync EntraConnect, we do not see that extra alias in the M365 admin panel (but no surprise, we see in local Exchange Hybrid).

Obviously, we still need to add their custom domain to our M365 tenant for the admin panel to "see" that domain, but the question is whether, after that point, adding these proxyAddresses through Set-ADUser will sync into EOL, or whether we have to do something else with EntraConnect, or whether we can just run a separate Set-Mailbox userMailbox -EmailAddresses @{add="[email protected]"} to match it up?

Separate note: I know we can add UPN suffixes in AD Domains and Trusts and sync the domain suffix that way, but this is purely for a mailbox alias, not identity; no one will be signing into AD or M365 with that domain as their UPN... https://learn.microsoft.com/en-us/answers/questions/781907/add-new-email-domain-to-office-365-hybrid

Thanks in advance, anyone who has done this!