You could spend a week in the Beartooths. Most lakes require a day hike in.
We usually just do Rock Creek and hike in a couple miles. Fish sizes small but very fun.
The first lake Shortyb showed was Greenough Lake at the base of the Hyway and it's perfect for kids. After you leave the Greenough - Lake Fork area and head up the hill, the next opportunities will require a Wyoming fishing license. You can buy one at the hardware store in Red Lodge or at the Top of the World Store near Beartooth Lake.
There's a couple of Lakes around the summit but when you see them down below the thought of hiking back up at 10,000 is a sobering thought.
The easy lakes to tackle are right along the highway as you come off the hill on the Wyoming side. It's Long Lake, right along the highway and Little Bear Lake. Towards the south you'll see Chain Lakes a pair of good size water usually shallow that also holds fish. There's a couple other lakes we hope to try soon only a couple mile hikes off the highway.
Got your 3wt, Little Bear Creek along the highway near the store holds lots of cooperative brookies.
Take the campground entry to Island Lake and when you crest the hill you'll see parking at the boat launch. Pretty nice lake you don't see from the road
and it's worth driving the 1/4 mile in for the view. The last one is Beartooth Lake. Nice picnic area at the boat ramp. Both Beartooth and Island Lake are very deep. Fish sizes also very good. We've had best early in the day or late evening working the edge of the deeper water.
I've also fished the first two pull offs on the Chief Joseph Hyw on the Clark's Fork of the Yellowstone down from the Beartooth Hyway Jct. Always do very well on 10-12" rainbows and occasional Cutthroats. There's a couple hikes near Crandall where the river goes into the canyon. The fly shops in Cody report larger cutties can be had but it's also very serious bear country. Have yet to try.
Since you have a little time you might want to buy a copy of Fishing the Beartooths by Pat Marcuson. It's the bible for this area.
Have fun and we'll expect a full report