We should remove every threat to those bighorns,...
I think that it's more a problem with the Park Service and their "keep it natural" attitude than it is an actual problem of goats and sheep competing for food.
In the late '70s until the mid '90s I would make several trips into Yellowstone NP each winter to see and photograph the bighorn sheep on their winter ranges on McMinn (sp?) Bench just south of Gardiner, MT and on the slopes just west of the confluence of Pebble Creek and the Lamar River in Yellowstone NP.
Thirty or so years ago the sheep on McMinn Bench were just about wiped out from an epidemic of "pink eye" that they contracted from the domestic sheep on the CUT ranch just northwest of Gardiner. The Park Service refused help from a sportsmen's group that volunteered to help trap and medicate the diseased bighorn sheep. The Park said it was a "natural" disease and that it should naturally run its course. The Park's solution was to put signs up along the road that read "No Admittance, Bighorn Sheep Management Area".
When the Park was preparing to introduce wolves into Yellowstone, one of the Park biologists said that wolves would not be a threat to the Bighorn sheep as "the sheep inhabited the steep, rocky cliffs where the wolves couldn't go". Total BS! I have pictures of a friend and myself sitting on the flat ground of the McMinn Bench sheep wintering area and a ewe walked up to me and sniffed my hand, and a ram walked up to within a few feet of me. Those sheep are extremely easy prey from wolves there.
Its also been many years since I've seen a sheep in the Pebble Creek area of Yellowstone NP. For the past 10 or so years the Park has posted this area as "No Admittance, Wolf Denning area.
For over 40 years and as recently as last summer I have enjoyed viewing mountain goats on Baronet Peak in Yellowstone NP, west of Cooke City. These goats are in the rocky cliffs where the wolves won't go, and I've never seen a Bighorn sheep near there. However this is another mountain goat herd that the Park Service has proposed to kill because the goats there "are not native, they moved in from the Beartooth Mountains in Montana", just north of there.
Like others have posted, there are plenty of mountain ranges in Montana that support healthy populations of both bighorn sheep and mountain goats.
I have hunted Bighorn sheep in at least 5 of the Montana Unlimited Bighorn sheep units (some are no longer Unlimited), and I shot a Mountain goat in one of those units. While hunting sheep, I have seen Mountain goats in several of the units, and one time I saw goats and sheep within several hundred yards of each other. Other than on the winter range, there is no shortage of food for either the Bighorn sheep or the Mountain goats. I have never seen Bighorn sheep and Mountain goats of the same winter range.