As a first pass, I would call the option "guest link" and have it within the Control Centre (in a similar location to the session link, but not directly next to it to avoid accidentally sharing the wrong one).
In terms of behavour, I'd take inspiration from paywalled news sites. The flow would be:
Open Control Centre and see an option "Create guest link (1 of 1 available)"
Click option and a sharing URL is created, and a brief explanation suitable for pasting into a mail/message is created. Perhaps something like:
Hi! I've been using Kagi for search, and thought you'd like to try it out! You can get temporary access without signing up using this link [link]. This will give you unlimited access for 3 days, after which you will need to create an account.
Once the link is used (as in, opened and a search performed) mark the link as used and subtract from the quota of guest links a user can create that month.
The guest user has something akin to a Session Link, tied to an account with no email address and 3 days of access. After the time is up, they will need to set an email address and pay to continue using the service.
Just to clarify 3 - the reason to avoid marking the link as used on first access is that various chat and mail services open and scan links in messages, which can break simpler guest acces mechanisms.
