The preliminary write-up for Entei already mentions this fwiw.The way I see it, Entei doesn't even really need a Time Flute, seeing how it makes incredibly good use of Shadow Rush. Hyper Mode effectively gives it a near-guaranteed crit with Shadow Rush, allowing Entei to make excellent use of its high Attack stat.
You can then purify Entei around the time it needs to level up to keep up with everything else (doesn't happen until Realgam Tower, since Pokemon in Colosseum are so weak).
You can easily get away with just having Skiploom as the only Sun teammate. Skiploom is already one of the best support mons in the game, has an effectively free moveslot for Sunny Day to set up Sun for an Entei partner (better than having Entei do it itself imo), and you can fill the rest of your team with Espeon/Ampharos/Flygon/filler without any type conflicts. Plus if you're going to grab Solarbeam you're getting Sunny Day for Skiploom, too.It's also the weakest of the legendary beasts in terms of synergy with the rest of the mons you can get.
It only really has Fire Blast after purification, with all the PP issues that entails. Natural Flamethrower is nice, but that's Lv. 51.
Solarbeam is nice, and Entei shines best in a Sun-oriented team, but the rest of the options for a team like that are... pretty grim especially since you do get so many better options for Rain.
Vibrava > Skarmory for the Time Flute. Vibrava needs access to evolution and Earthquake ASAP to be good, while Skarmory doesn't have to be purified to do its job of sitting there and eating hits.I'd argue that, assuming you're using these mons, of course, the Time Flutes should almost always go to Suicune, Raikou, and the last one is kind of a toss up. I'd personally consider Skarmory, but I can see people wanting to save it for Metagross or TTar.