Weather needs to be a consideration at the end of May. Rain is likely on a northern route. Heat is likely on a southern route.
Let me see if I understand this. If you're leaving NY on 5/28 and flying home on 6/9, then you're going to ship bikes from SD to NY in early June. (I originally suspected the reverse.)
You will probably need to have the bikes at the shipper on 6/6 or fly late 6/9. This means your total trip, coast to coast will be effectively 10 to 13 days. If we figure googlemaps' 2804 miles, that's a pretty short haul each day.....less than 300 miles on a straight shot.
So, my suggestion would be to start in the NYC area. Head thru PA's Amish country, then southward toward the VA, WV, then the Blue Ridge mountains. Try to get "the Dragon" into the mix. Continue south....stay on the beach in Pensacola, FL (southern tip of Alabama).
Then visit the west..... I-10 and I-20 in Texas are the most boring roads on the planet, but once you get into AZ, the views can be fantastic. From there, it's a straight shot into SD.
But then, there are the other routes. Just remember, it gets boring across the middle of the country, wherever you decide to roam. North, central, south..... Kansas is a snoozer. The Dakotas are snoozers.
Ask your buddies if there's anything they especially wanna see. Yellowstone? California's Pacific Coast Hiway? Plan some of your trip around a goal or destination. If this is the epic trip, you should do what you want and see what you want.