You are correct about splitting into many ways using the splitways tool. Check all that the ways are not ‘over-noded’, using the JOSM Simplify Way tool.

For tagging, the ways would contain no tags, but would all be members of a multipolygon relation.

type=multipolygon
boundary=national_park
leisure=park
name=Alpine National Park
wikipedia=Alpine National Park
website=

Each way would have a role of “outer” in the relation.