On the whole though I believe transitioning to a 3-4-3/3-4-2-1 formation would alievate some of the problems. We'd have that extra man in defence whilst our fullbacks - now wingbacks - would still be able to cover the flanks, whilst retaining the front three he seems to want to persist with.
CB CB CB
WB CM CM WB
RF CF LF
I think that approach would work, but only if Klopp drastically alters his approach to football and asks his team to sit back and only play on the counterattack. While I fully agree it makes us more solid when we are in our defensive shape, we'd get exposed in midfield even more if we'd still try to dominate games like we are doing now. As of now we're struggling to track runners with three midfielders. The solution then isn't taking one out and replacing him by a defender who plays further back.
We've actually seen Klopp do this a handful times this season when we're ahead late in games. He'd bring on a 3rd CB and play 3-4-3 or 5-2-3 whatever formation you want to call it. This is when we're under pressure and sitting back, where I think the formation would be useful.
Our attacking shape actually isn't too far off a 3-4-3 anyway if you look at it:
On the left is our current formation on the opposition half. That looks a lot like a 3-4-3. On the right is the formation you suggest when we're attacking (correct me if read it wrong in your second post.) When we dominate play, you see Henderson dropping in between the CB's, effectively becoming a '3rd CB' to either initiate the attack from deep or to keep things ticking over. In your formation, that would be an extra defender. We would then lose a big part of our ability to win back the ball in midfield and retain possession if you want your 3 CB's to hold their line. If you don't want them to hold their line, there's no point in playing the 3 and you might aswell play a midfielder there. We would also lose a man when playing the ball around the opposition box trying to create gaps in their defence (Usually we'd have Moreno - Wijnaldum - Can - Trent) In the 3-4-3 that leaves Moreno - Wijnaldum - Trent.
Harry also makes a good point:
Another big question is how this change would affect our ability to press in unison. Our best 11 is pretty well drilled at the high press, but a shift like this would create a lot of organizational problems for us as to where and when we initiate the press.
If you look at the formation, I worry about the zones on the field I marked red. While far up the pitch we might not have any problems with our pressing in your 3-4-3, if the opposition teams manage to get the ball in the red areas, we would have problems to effectively press the ball.
If for example in our current formation the ball is played in the left red zone. Everything shifts to the left. We immediately have Moreno, Wijnaldum and Henderson counterpressing the player with the ball while, in this case Can, drops deeper a bit to make sure he fills the gap in midfield and cuts off passing lanes into midfield, avoiding the opposition team to be able to easily switch play from one side of the field to the other.
In your formation, shifting left to press the player with the ball, would leave a massive gap in midfield, in front of the 3 CB's. If the player then manages to play the ball into that zone and their midfield runners break forward, our 3 CB's are on the backfoot with the opposition team breaking forward with acres of space to run in.
An other worry, we have actually given teams a blueprint of how to play against a 3-4-3. We tore Arsenal and Hoffenheim a new one and they both used this formation or a variation on this formation. Because we're leaving more players behind the ball, we allow the opposition team to leave more players up the pitch (think Mane against Hoffenheim). If they would leave players in the zones I marked and just one striker in front of the 3 CB's, we would be in trouble. They would try to create the situations we tried to create in these two games too (and actually against City aswell with 11 vs 11, Otamendi vs Salah). I don't want to see our defenders high up the pitch or having to run backwards in 1 vs 1 situations. Hoffenheim managed to create a few (Gnabry against Lovren and that resulted in a penalty). As other posters have pointed out, we don't have the midfielders to play that formation. Any team that would play us, would make our midfield look like we made Xhaka and Ramsey look against us.