Doing a transcon in 30-45 days would be a little harsh; it's doable, though (see
www.pactour.com -- $3,295), but wouldn't leave you all that much time to stop and look at things.
Take a look at Adventure Cycling (
http://www.adventurecycling.org/tours/index.cfm) for some camping-based (and therefore, not insanely expensive) tours.
Given a 6-week limit, the Lewis and Clark and Northwest Coast (30 days -- $1499), fit both in time and budget.
Both of these would allow you to get a very nice bike.