Article III, Section 2 of The Constitution requires that
The trial of all crimes, except in cases of impeachment, shall be by
jury; and such trial shall be held in the state where the said crimes
shall have been committed; but when not committed within any state,
the trial shall be at such place or places as the Congress may by law
Since the Idaho portion of Yellowstone is part of the state of Idaho and is not part of a non-state, ergo the trial shall be held in Idaho, according to The Constitution.
One jurisdictional statute, 16 USC 24, says
The Yellowstone National Park, as its boundaries now are defined, or
as they may be hereafter defined or extended, shall be under the sole
and exclusive jurisdiction of the United States. All the laws
applicable to places under the sole and exclusive jurisdiction of the
United States, shall have force and effect in said park. Nothing in
this Act shall be construed to forbid the service in the park of any
civil or criminal process of any court having jurisdiction in the
States of Idaho, Montana, and Wyoming. All fugitives from justice
taking refuge in said park shall be subject to the same laws as
refugees from justice found in the State of Wyoming.
So the crime would be a federal matter. I put emphasis on a sentence that doesn't have a straightforward interpretation. There is another jurisdictional statute 28 USC 131, the problematic one, which states:
Wyoming and those portions of Yellowstone National Park situated in
Montana and Idaho constitute one judicial district.
Court shall be held at Casper, Cheyenne, Evanston, Lander, Jackson,
Therefore, the crime must be tried in Wyoming. Moreover, the judicial district of Wyoming also includes that unoccupied portion of the park in Idaho. This statute conflicts with Article III, Section 2, which requires court to be held in Idaho. Since The Constitution has precedence over federal statutes, the instruction to hold court in Wyoming is unconstitutional.
The 6th Amendment does not pose a particular problem, with its requirement that
the accused shall enjoy the right to a speedy and public trial, by an
impartial jury of the state and district wherein the crime shall have
been committed, which district shall have been previously ascertained
since the jury could be of Idahoans, and transported to Wyoming – the 6th addresses the nature of the jury, not the location of the trial.
It is an open question what the 6th Amendment refers to in talking of "state and district", since district is not defined at a constitutional level. The only district with constitutional stature is the District of Columbia, which is not a state, and therefore there could be no "state and district" if we mean "district, as specifically mentioned in The Constitution". The Judiciciary Act of 1789 first created federal judiciary districts, signed by Washington on September 24, 1789, and the 6th Amendment was passed by Congress the next day. It is reasonable to assume that this is what they meant by "district".
28 USC 131 does say that all of Wyoming plus a sliver of Idaho are a judicial district, so one could have a jury composed of citizens of Wyoming which satisfies the "jury of the district" requirement, but that would not satisfy the "jury of the state" requirement. If anyone lived in that area, those people could constitute a "jury of the state and district", but you would still have a clash between the statutory and constitutional requirements for venue.