In Toolset Views, searches will not respect Toolset Access settings, this is a known "issue", but it's intended because the query just would be crashing too many use-cases.
Hence, a View should be shown/hidden in its entirety, if not results shouldn't be visible, or using Custom HTML Conditions, or similar approaches to hide/show content.
That should not be different in Relevannsis Search enhanced searches as the basis is still the same query (WP Query).
I see you found API provided by Relevannsi so to control what it outputs, which should be the correct way to go.
Toolset Access cannot control output in those queries, for performance reasons.
That said, you can suggest the feature to the Product Management, it is always possible, that our last findings are eventually not appropriate anymore, however it would require a new Feature (to make Access Rules respected in any query run on a site).
Please let me know if you need help adapting your Custom Code filter to listen to Toolset Access set rules, I can help with this . because right now, your code only would listen to password protection, which is not set by Access.
In case you need help with this, I will likely need access to a staging site or a very detailed step by step description of your rules applied in Toolset Access, and Relevannsi search, so I could replicate and provide the right solution to listen to Access settings.