This would work way better for your timeframe and plans.
Three months is a liberal timeframe for a cross-country tour.
At 4000 miles, that's fewer than 50 miles per day -
Fewer than 60 with a day off per week.
Working backwards from the west coast:
Three weeks in August to visit and tour - so you arrive Aug 1.
The month of July to tour from Yellowstone to Glacier via US 89 to Anacortes via the Northern Tier.
(Possible loop down to Missoula - if Missoula is also a definite.)
Do not miss Glacier in early to mid July if you are heading to the Pacific Northwest.
Where do you want to cross the Great Plains? I suggest Nebraska over Kansas.
There is also the possibility of taking in the Black Hills of South Dakota.
That might suggest St Louis to Rulo,, Nebraska via the Lewis & Clark -
NE 2 west of Grand Island is pleasant, NE 92 has stunning grasslands landscapes.
Since you have plenty of time, I would suggest including the Black Hills and Bighorn Mtns.
Hot Springs, Spearfish, Devils Tower, Buffalo, Cody to Yellowstone.
Even if you spend a week each in St. Louis, Lexington and Baltimore -
I can't see how you would need to start before April 15 - - or want to.
By staying near the coast you have a few extra degrees.
How about the Cape May ferry - then cross the Chesapeake at the Lane Bridge?
(You would need a shuttle to get across the bridge - your friends?) Then ride up to Baltimore.
Crossing the Appalachians in late April/first of May will be chilly and wet.
Consider using the North Bend Trail in WV then following the Ohio River to Huntington.
I have biked from St Louis to Louisville to Lexington - some great back roads.
The weather should be ideal for much of this trip except at the beginning.
If you are flexible about the start date - i.e. willing to wait a few extra days -
You can minimize the possibility of a chilly, wet, nasty start to your adventure.