What is the best way to fit the sites I want to see in Rome into two days?

I remember having walked one evening from Vatican to Termini train station and found my way through Piazza di Spagna and Fontana di Trevi. Maybe Piazza Navona as well.

You definitivelly should discover Roma by foot. The city is not that big.

By the way, one full day in Vatican is what I would have suggested if you hadn't mentioned your schedule. The view from the top of St Peter is worth the hundreds steps to get there.

Your aggressive sightseeing goals are entirely possible. We were in Rome when our cruise ship docked in Civitavecchia, which is about 1 1/2 hours from the city center! We were able to finish all these spots (although the Sistine Chapel was closed), with a few 60+ year olds in tow. That said, it was extremely tiring, and I wouldn't do it again the same way.

Your day breakdowns seem well organized - and all are within walking distances, once you get to the general vicinity. For the Colosseum, ensure you make it there first and early, otherwise the crowds will get heavy.

Make sure you are staying in a centralized location. For transportation, purchase a 3-day pass which should allow you to get around easily. Be prepared for hordes of crowds!

You've pretty much sorted things out between Day 1 and Day 2. The only observation I'd make about Day 1 is that the two non Roman sights "book end" the Roman sights south and north. Starting in the south, and moving north, you would see the Spanish steps, Colosseum, Roman Forum, Pantheon, and Trevi Fountain (nearest the Vatican) in that order. If you wanted to go north to south, then reverse the order.

The second observation is that each "day" contains TWO days worth of sightseeing. Trying to cram four days into two, you'll be tired by the end of the second day (assuming you were reasonably fresh when you arrived), or exhausted, if you arrived tired. Even so, if that's all the time you have, better to see it "all" and be tired.

Most of the distances are close enough to walk. Depending on where your hotel is, you may need transportation to and from your start and/or end points on one or both days.

I would add Piazza Novona and Palentine hills in Day 1.

Piazza Novona Fountain of the Four Rivers, one of Rome’s most iconic fountains located in the square of the Piazza Navona. It was unveiled to the public in 1651, over 400 years ago.

It is usually bustling with artists showcasing their talents.

I would not miss Borghese Gallery
You can do it on Day2.