I'm not a Montana native, but I've been to the Beartooths several times and have noticed that there seems to be much less snow east of Beartooth Pass than west of it, so even if there's little snow at 9000' around Red Lodge, things may be very different around Cooke City.
For example, last year I backpacked up to Black Canyon Lake on June 29 and climbed two peaks out there. There was very little snow getting to the lake (see first photo below) and little at the lake (9200'-- see second photo), but there was plenty above 10,000'.
On July 1, however, I packed in to the Goose Lake and Grasshopper Glacier area after finding the 4wd road blocked by snow about a mile in and only a little over 8000'. Almost the entire hike was a slog through deep, soft snow. It was still winter out around the lake, as seen below, and the lake is at 9600'.
From the summits, I could see that the entire area was still buried, to the Aero Lakes and well beyond.
You'll be there three weeks later than I was, so you have that going for you, but this was also a very heavy year by all accounts. Unless it gets hot soon and stays hot, you may be contending with a lot of snow cover still. My experience from last year tells me that it will be nice and mushy, so get ready for postholing that will really slow you down. That would be my concern more than following the route, which should be fairly easy given a good map, compass skills, and cooperative weather.
I hope it goes well and that the conditions are much better for you than they were for me.