Search Results API call fails with any URL params

I’m not sure if I understand your question correctly, but the following URL works on my test site:

https://swelteringaunt.s2-tastewp.com/wp-json/wp/v2/search?per_page=5

OK, found this in the documentation:

If you are using non-pretty permalinks, you should pass the REST API route as a query string parameter. The route https://oursite.com/wp-json/ in the example above would hence be https://oursite.com/?rest_route=/.

So /?rest_route=/wp/v2/search is for non-pretty permalink.

Since we have a query parameter in there already, to add a second query parameter, we need to use & rather than ?, so these two queries are identical and both work:

With Pretty Permalink:
https://swelteringaunt.s2-tastewp.com/wp-json/wp/v2/search?per_page=5

Without Pretty Permalink:
https://swelteringaunt.s2-tastewp.com/wp-json/?rest_route=/wp/v2/search&per_page=5