You can create a filter and apply it to the filters properties of the embed configuration. Check Embed Configuration Details.
The problem now is, there's a bug that stops the filters properties from working, check validateReportLoad incorrectly returns schema errors for valid filters. The workaround is, as your mentioned, set the filter after loaded.
Regarding how to set the level(Report/Page), check Understanding the object hierarchy, Both Report and Page have the setfilters function.