Branch, Tag and Organization searching cabilities with GitHub APIs

I have a requirement to search for branches, tags and organizations by providing a search text such as branch name etc. through the GitHub API.
But I was not able to find anyway of achieving this.

There is an API for search : https://docs.github.com/en/rest/reference/search, but this can only seach repositories, commits and issues.

I also looked in GitHub GraphQL endpoints to see whether they support searching mechanism but it seems not!
GraphQL search : https://docs.github.com/en/graphql/reference/queries#searchresultitemconnection

I also found following community thread on branch searching and it also concluded with a negative answer.
https://github.community/t/api-filter-search-branches-by-name/13565

I have large number of branches (over 1000) and repositories (over 500) within my organization. Therefore, taking all paginated results and implementing client side search mechanism is not a feasible option.

I want to know what are the options we have for searching branches, tags and organizations.