Do you think this is a feasible 3-night itinerary for Rome?

Read the Rick Steves Travel tips - NEVER rest on arrival . . . Push through and go to bed after dinner. It is tough - but always recommended to do it this way. Drop off your bags, grab some food and WALK & explore.

The Colosseum and Forum are best seen and experienced during daylight. They are next to each other, so do them together.

I'd also add San Clemente to your Colosseum/Forum day - it is so overlooked. It is an incredible church built on 4 levels of history - going all the way to the 4th and 1st centuries. Hire Private Guides for everything - you will see and learn so much more.

Probably will arrive in the morning. Here's the rough plan:
Day 1:
- Day: Rest at hotel
- Late afternoon: Look for TIM SIM card

If you can do early check in, it is best to do not more than a 45 minute nap and then get out and keep moving.

How many are you trying to house for $250 per night? (How many is 'we'?)

Just two people.

I've heard that you shouldn't rest upon arrival but it's just not realistic for me. I struggle with long transatlantic flights. There would have to be a short nap of at least 20 minutes. I just don't see myself doing any "major" activity in the morning/afternoon of the first day. It's never been done before, so there's no reason to think it'll be done this time.

If you Google around, some people think the Colosseum is better at night. The lighting scheme is supposed to be nice and we don't really like crowds.

Edit: Here are some hotels I've come across, in case you can recommend one based on personal experience. Hotel Damaso, Arch Rome Suites, Hotel Barrett, and Little Queen Suite.

We have almost always been able to get into our room on arrival from the East Coast. We have never been able to sleep on planes.
We do take a short nap, then often a quick shower and head out for the late afternoon and evening- staying outside, lots of walking. To bed after dinner
We do not suffer any further jet lag, works for us.
Do what works best for you.

I would suggest just doing the RS Heart of Rome walk on arrival- Trevi, Spanish Steps, P Navonna- you can do it in either direction. Stop for drinks then dinner and then gelato.

I would put Colosseum with the Forum etc just because al in same area. You might even put Pantheon on that first day- you walk right past it.

we don't really like crowds.

Who does? Just realize that there will be crowds pretty much everywhere all day and night!

Evening Day 2- consider a food tour-tons of fun. EatingEurope.com (but not sure after Vatican AND St Peter's --that's an exhausting day ;)

Use Booking.com to search for lodging- use the filters for budget, location, amenities like AC, etc
We like anywhere in the historic center- near Pantheon, P Navonna, Campo de'Fiori- can walk everywhere from there
Read the reviews there as those can only be left by people who have actually stayed at the property. You can book there or find the hotel website- see who offers the best deal.

ETA-

Hotel Damaso

very nice, great location

Another nearby is Hotel Smeraldo- a forum favorite

TIM has a store in the baggage area of terminal 3 at FCO. No need to go hunting in town.

hey hey reeddj
you are traveling during high season and it will probably be HOT with crowds up the ying yang with the places/attractions you are planning to see.
check-in is 3-4pm and check-out 10-11am unless you get early or reserve room night before and let hotel know you plans. not anything new it's what hotels are like all over. drop you luggage off until check in and walk around. many people want to travel early morning then wait or rooms available till check-in. i've seen it happen and they are not happy, can get nasty/grumpy with workers, they just work there, read check in rules.
you are not spending that much time in rome, but do reserve early/ASAP for vatican/sistine chapel and colosseum/pantheon at direct website, tickets go so fast. have small coins to throw in trevi fountain, many places are charging fees, do your research for all your activities. lines will be long, you will have to wait in line for security like every one else, restaurants will be busy in tourists places, many want reservations. we use to walk down small alleys/calles/street for small mom & pops away from crowds.
florence is another crazy/crowded place, prices will be high will supply and demand, many have booked months ago with your budget low when centrally located,
booking.com
put your filters on left side, get A/C and see what they offer, any mobility issues climbing steps, cobblestone/uneven pavement/hilly areas, cancellation policies, kinda late with reserving. good luck hope you find what you want
aloha