You could serve your API over different names : api-basic.myapp.com (that one does HTTP basic auth) api-oauth.myapp.com (that one for OAuth2) api-whatever.myapp.com (and the sky is the limit...)

You then point each type of client to the proper name.

These names can point to the same endpoint which will behave differently according to the name (ex: kubernetes ingress) or even point to different endpoints completely (an ingress on IP No 1 and a separate Nginx reverse proxy on IP No 2 for example).