Thanks. It sounds like you are quite flexible and basically willing to do whatever. Since you are a total novice, here are some suggestions.
Take the ACA's Chicago to NYC route (in reverse) from NYC to Denver, IN. From there, connect to the Northern Tier route to Muscatine, IA. From there, I can get you a route to Sioux Falls, SD, where you join the Lewis & Clark route until you can connect with the just released (this week!) Parks, Peaks, & Praire route to connect to the classic TransAm route to Kooskia, ID, where you connect again with the L&C route to Astoria/Portland.
Since you are a novice, I highly recommend the ACA maps for your first adventure. Though a tad pricey, they are very good maps and list all the services a cyclist needs. You can always sell them on eBay afterwards for about 1/2 price.
I would caution you to plan for no more than 60 miles on average per day. If you have only 45-days, that would NOT get you across the country. More than 60 miles per day on average (factors in rest days due to rest/sight seeing, weather, mechanical, etc.) and you are actually riding 70+ mile days which is a pain especially when you are camping. The 65-70 mile per day may not sound like much but it really adds up and takes away the time you can tour (stop and smell the roses) versus just plowing through. The ACA maps help speed your time along as they have already located the hotels and such for you. Do not underestimate the time it takes to find a campground, etc. It can easily add an 30 minutes or so to your day.
Another option but a hot one at this late date is to go from Brownsville, TX (or Houston) to say Winnipeg. Again, I can help with a route as I have ridden this route. The big plus is that you could do the route in 5 weeks as you will get tailwinds most days. The big negative is it will be HOT, especially in south Texas.
A third option would be to take the train down to Miami and head north on the Atlantic Coast route. This would meet your timeframe but it would be hot and humid until you got close to home. A HUGE benefit is that you are not rushed to make a flight, train, etc. I can not tell you how much that can ruin a trip when you are on a schedule to catch the flight.
Honestly, if I were you knowing what I know after 40+ years of touring is to take the train out to Portland, start the TransAm route and take that to West Yellowstone and jump on the new PPP route to Minneapolis where you can catch a train home. The route would fantastic scenery wise, gives enough history to enjoy, and you are not rushed. Additionally, the weather would be more acceptable though the plains would be toasty at times.
Tailwinds, John