This question is a no-brainer. Follow the TransAm.
I have to agree. Riding the ACA Trans America route is a great way to go. Similarly the Northern Tier might be a good option. There is some chance for flexibility by combining other routes like the Lewis and Clark or the Western Express with portions of the Trans America, but personally I took the TA as my first choice and probably would again. We did improvise a few sections, but mostly stayed with and liked the route very well.
I think that the listing of places to stay (including many free or cheap ones) and other services like emergency numbers, bike shops, post offices, libraries, and so on are a real asset.
I also think they did a great job of picking the route. Which picks a lot of country roads and small towns and not too many cities.
There is a nice amount of other bike tourists on the route. Enough so you can meet and make friends along the way but not so many that it is objectionable. You will probably meet someone else riding the route every few days or so depending on the date and where on the route you are. You will be able to camp with other tourists once in a while and will also be able to be alone as much as you want.
Your projected time of two months works out to averaging 70 miles per day if doing the TA. That is do-able, but I would advise an open ended finish time with the option of taking longer if possible. It is nice to not be tied to a schedule.
Starting in June in the West is probably OK, but later in June is better than early if starting in the West. There would be a good chance for McKenzie Pass to still be closed Early June (you can detour on Santiam Pass, but it would be a shame to miss McKenzie). Starting in the East, earlier than June is better weather wise, missing the heat and humidity of the SE and hitting the west when the snow is reliably gone on the passes. That said June is OK.