I hate to defend Biden on anything but wouldn’t housing the Vice President count as an “official use,” the same as staying at the Naval Observatory or Camp David (or riding on Marine 1)?
Under park service rules, the lodge is maintained for use by federal employees for “training and official conferences” and for those on “temporary duty in the park.” In practice, the superintendent of Grand Teton National Park, who has discretion over whether to demand payment for the lodge, has interpreted those rules to allow extended family vacations if there is an element of official business involved.
I'll wait to see what official business is claimed.