I agree with the advice you have received so far. That is a lot to do in one day. Although all the sites are next to each other on the map it takes a while to walk through them all. Remember you gotta eat! If you feel bound and determined to see all of these sites in one day then get up early start at one end and work your way across to the other. Most sites open at 8:30. However, if you plan to see everything on your list you will not have time to go into every site. You could take photos of the Victor Emmanual Monument (in the Piazza Venezia), the Piazza del Campidolglio (on the Capitoline Hill), Imperial Forum and Trajan's column before the other sites open at 8:30am, (do not go in the museum or up to the observation deck of the V.E. Monument, you likely don't have time). You could try to fit in a visit to the Pantheon, not sure why it is less important to you than the other basilicas but incase it was an oversight on your part.... It is free, can be done quick and opens at 8:30. Next head to the Palatine Hill to buy your pass, see that area first, the Forum second and then break for lunch. Next The Basilica di San Pietro In Vincoli. After that continue down Via del Fori Imperial with Imperial Forum ruins and the Parco del Colle Oppio on your left until you get to Basilica of San Clemente. Finally, finish your day at the Colosseum who's crowds start to thin around 3pm.
This will be a very intense day and you will be tired! You may get to a certain point and throw in the towel, but in theory it could be done this way. If you get too hot or want to give up walking around in the heat, the museums on the Capitoline Hill are air-conditioned. However, if you go into those museums you will be forfeiting time to see other sites on your list. Most travelers have a saying "I'll get it on the next trip" and then we make sure we go back. Be willing to try this out, it is impossible to see everything cool in and around Rome in only 2 days. Buena Fortuna!