Both are good alternatives. The direct WFS request should include a Contains filter, or with point-in-polygon search the Intersects filter is as usable.
You can even use GET instead of POST in the request but URL-encoding the XML filter makes it not look a bit ugly
Be prepared to spend some time with testing the right order of lat and lon coordinates if you start making filters because unfortunately that depends on your target server, the coordinate reference systen, and WFS version.
Using ID from GetFeatureInfo may feel double trouble but on the other hand WFS version 2.0 defines an mandatory stored query GetFeatureById (some more information in https://docs.geoserver.org/stable/en/user/services/wfs/reference.html).
The syntax of the GetFeatureById is rather simple, here an example taken from the ESRI site https://enterprise.arcgis.com/en/server/latest/publish-services/windows/communicating-with-a-wfs-service-in-a-web-browser.htm