@Davis, Vincent , Based on my understanding of your scenario description, you may try using a custom analyzer at index time that removes or replaces special characters in your indexed fields. Tutorial: create a custom analyzer - Azure AI Search | Microsoft Learn.
If you haven't tried this already, you may use a character filter that removes or replaces the "$"
You may also use regex, something like this: "search": "/\\$+Benson, Carl/"
Kindly let us know how it goes I'll follow-up with you further.