Possibly, though the Rad Libs were the party that was more in favor of peaceful relations with the U.S., so TR might think they would make for more trustworthy "partners" (i.e. puppets) than the Whigs.
If TR has has gone to the time and trouble of kicking a sitting CS President out of office, I'm not sure he'd bother with mere puppetry; he might well demand a full-blown military occupation of the Confederacy and it's entirely possible that demand would do more damage to his position than the Confederate States of America circa AD 1922 ever could (Especially in the year of mid-term elections).
Hngh, that last bit actually explains why the old Bull Moose might bother himself with appointing a puppet government in the first place; truly the ballot box is more fearsome than the bomb in a politician's imagination!
It may ultimately come down to how exactly Featherston gets overthrown.
For my money TR in power means that Jake Featherston never gets any nearer the 'Grey House' than screaming abuse from the front garden - I'm morally certain that even some Stalwarts would think twice about backing anybody mad enough to talk smack to Theodore By-Good Roosevelt at the peak of his powers, so the 'Weathervane Whigs' are going to run screaming (and I remain certain that it was Whig opportunists looking for a better offer, quite as much as the angry floating voter, who handed Jake Featherston his shot at World-class infamy).
(President Ainsworth Layne perhaps.)
It's rather amusing to wonder if AL would actually
accept the Presidency under such circumstances; I've done a bit of thinking about him as a character and I rather love the idea that he's entirely capable of telling TR "No" to his face and sticking with that resolve in the face of pressure to change his mind (Eyeing the nearest exists & shivering like a jelly in an earthquake all the while).
My mental image of the character is, you see, an interesting compound of High Principle and sensible cowardice (He'll scream, he'll thrash, he'll run for cover and beg for mercy but the little ****** just will not tailor his principles to suit the situation*).
*
Hence his making no reappearance in the series; like Doroteo Arango he's something of a one-off, though unlike Señor Arango he's more of a Jimmy Carter than a Pancho Villa (Which is to say that he's an admirable progressive who'd make an absolute hash of the politics, but probably accumulate a much better reputation post-Presidency through consistent Humanitarian work - especially if the Radical Liberals are able elect more successful Chief Executive, allowing them to regard Layne as a version 1.0 rather than a disastrous one-off).
Assuming Featherston actually survives getting overthrown (which knowing the Snake isn't too likely as he also doesn't seem the type "to go quietly), I imagine the U.S. would insist on exiling him and his top cronies to whatever the U.S. equivalent of St. Helena is.
Almost certainly that former Leper colony in Hawaii, although the Queen Charlotte Islands might also merit consideration; the Aleutians might be a little too far out (and are more or less Russian in the 1920s, unless TR decides to double down on Canada's lifetime supply of ice, snow & angry non-US Americans).
p.s. For the record, when I was referring to Featherston being arrested & imprisoned it was in terms of a timeline where he does not enter the Grey House in 1921.
A Knight vs Long presidential election somewhere down the line would certainly be interesting.
It's interesting to wonder where Ferdinand Koenig would come down in a timeline where Featherston was swept out of Confederate politics - that is, I suspect, as interesting a thought to ponder as would be the question of who win in what is likely to be one of the more knock-down, drag-out, no-holds-barred elections in Confederate political history.