I'll join the chorus of calls to skip the Black Hills, save your time for the GYE (and UT & CO).
Other options to check out just north and west of YNP: Mt Cowen and Hilgard Peak
Both could be done in 2 days if necessary and offer good challenges (Class 3-4) and stellar views without the crowds, excluding occasional crowds at Elbow Lake with trad climbers scaling the local granite (not the main route) on Cowen. Could be good backup options in case you can't get the backcountry sites for the YNP trips as well.
Make sure to time your trip appropriately, which could be difficult with your itinerary. For instance, the SW couloir on Granite was socked in pretty late this year, as discussed here
. The standard route on Gannett can be more difficult if the bergschrund opens up in late summer (don't know when/if that happened this year).
FYI: Current projections suggest another year of above average snowpack in the northern Rockies.