First, a couple notes: sparing Yuthura on Korriban cancels out the 2x DSP you get from one of the quests in the Academy, so as long as you enter LS aligned and don't do both of them, you're good; also, the Republic soldiers by Zelka actually only give one increment of LSP, but fortunately that is irrelevant.
The Republic soldiers take about 15 seconds, assuming you're visiting Zelka anyway (about 20 seconds on its own), though if the door takes a while to bash in, it can take a bit longer. Also, if you wanted a second weapon to help with the door, your best option would be to equip long and short swords when you equip armor on the Endar Spire.
Getting LSP by giving the Undercity beggars 20 credits amusingly does not cost any real time; both ways take about 15 seconds. However, the several seconds saving and loading aren't timed, so it's still a bit of a loss. That said, if you messed up selling things and didn't have enough to pay the bullied merchant or just want more LSP to be safe, this is the place to get it, because it means fewer quicksaves and therefore less memory leakage.
Saving the Duros in the Sith Base is also really fast. It can be done while your AI-controlled party kills the two droids in the room for a comparable time loss to the one above.
So on to Korriban...
The doubting Sith takes about 10 seconds, and interrogating the Mandalorian about 40. A force jump or two could be possible on the interrogator after returning from outside, but it's still far slower. Unfortunately, it's necessary to leave the Academy in order to get enough prestige, because the DSP from both quests in the Academy could be made irrelevant by getting the two extra LSP on Taris, and going outside means a lot of running. So that basically means the fastest prestige route is (in no particular order):
1: Sith Code
2: Doubting Sith
3: Tattling on Yuthura
4: Holocron
5: Admitting you killed Lashowe
One other note I'd like to make is that the trigger to get the [Persuade/Lie] option with Yuthura is pretty weird. I think it might be that you have to be LS at a certain point before activating the ability to talk to her about getting into the Academy, but I honestly don't know. Just now, I did a test where I had plenty of Persuade ranks but was one shy of LS when I got to that point, then got LS by running back towards the Ebon Hawk and bribing the aggressive Sith with drinks, and only got the regular [Persuade] option. Generally it shows up when you follow the route, but unless one of us can figure that one out for sure I'm inclined to take the safety LSP in Taris Undercity.
Finally, based on my experience with the Persuade checks I'm pretty sure that Yuthura and Lashowe (or at least the second one with her) are both "Hard" checks. I would occasionally fail both of them at level 7 with my 10-11 Persuade RPGLB build (10/11 and 11/12 ratios, so only hard checks could fail, according to the mechanics stated). With Scout, the negative Charisma modifier will be canceled out by taking Empathy, so putting the maximum number of ranks in Persuade would give about a 75% success rate (9/11 or 10/12 ratio) - though because just how things are handled beyond the ratio isn't stated, I can't give a precise number. On the other hand, the 7 ranks you could afford if you maxed Treat Injury would yield a ratio that would give closer to a 25% success rate, which again seems about right based on my testing.
But then it gets weird. Either Valor has no effect, or FLU messes things up, because even with Valor my RPGLB build could fail - and Valor would up the ratio to 1/1 for both checks. I have some suspicion that it goes away after affecting one check, assuming it has an effect, but I tested with Yuthura and with her there only is one check. Much like chain levels in FF12, I think the known/stated mechanics may not be quite how things work...
tl;dr Persuade is weird, and it's worth maxing it going into Korriban
The Republic soldiers take about 15 seconds, assuming you're visiting Zelka anyway (about 20 seconds on its own), though if the door takes a while to bash in, it can take a bit longer. Also, if you wanted a second weapon to help with the door, your best option would be to equip long and short swords when you equip armor on the Endar Spire.
Getting LSP by giving the Undercity beggars 20 credits amusingly does not cost any real time; both ways take about 15 seconds. However, the several seconds saving and loading aren't timed, so it's still a bit of a loss. That said, if you messed up selling things and didn't have enough to pay the bullied merchant or just want more LSP to be safe, this is the place to get it, because it means fewer quicksaves and therefore less memory leakage.
Saving the Duros in the Sith Base is also really fast. It can be done while your AI-controlled party kills the two droids in the room for a comparable time loss to the one above.
So on to Korriban...
The doubting Sith takes about 10 seconds, and interrogating the Mandalorian about 40. A force jump or two could be possible on the interrogator after returning from outside, but it's still far slower. Unfortunately, it's necessary to leave the Academy in order to get enough prestige, because the DSP from both quests in the Academy could be made irrelevant by getting the two extra LSP on Taris, and going outside means a lot of running. So that basically means the fastest prestige route is (in no particular order):
1: Sith Code
2: Doubting Sith
3: Tattling on Yuthura
4: Holocron
5: Admitting you killed Lashowe
One other note I'd like to make is that the trigger to get the [Persuade/Lie] option with Yuthura is pretty weird. I think it might be that you have to be LS at a certain point before activating the ability to talk to her about getting into the Academy, but I honestly don't know. Just now, I did a test where I had plenty of Persuade ranks but was one shy of LS when I got to that point, then got LS by running back towards the Ebon Hawk and bribing the aggressive Sith with drinks, and only got the regular [Persuade] option. Generally it shows up when you follow the route, but unless one of us can figure that one out for sure I'm inclined to take the safety LSP in Taris Undercity.
Finally, based on my experience with the Persuade checks I'm pretty sure that Yuthura and Lashowe (or at least the second one with her) are both "Hard" checks. I would occasionally fail both of them at level 7 with my 10-11 Persuade RPGLB build (10/11 and 11/12 ratios, so only hard checks could fail, according to the mechanics stated). With Scout, the negative Charisma modifier will be canceled out by taking Empathy, so putting the maximum number of ranks in Persuade would give about a 75% success rate (9/11 or 10/12 ratio) - though because just how things are handled beyond the ratio isn't stated, I can't give a precise number. On the other hand, the 7 ranks you could afford if you maxed Treat Injury would yield a ratio that would give closer to a 25% success rate, which again seems about right based on my testing.
But then it gets weird. Either Valor has no effect, or FLU messes things up, because even with Valor my RPGLB build could fail - and Valor would up the ratio to 1/1 for both checks. I have some suspicion that it goes away after affecting one check, assuming it has an effect, but I tested with Yuthura and with her there only is one check. Much like chain levels in FF12, I think the known/stated mechanics may not be quite how things work...
tl;dr Persuade is weird, and it's worth maxing it going into Korriban