Rabbi Shimon Bar Lakish is known to have been a bandit before repenting and becoming a Rabbi.
Banditry involves thievery and violence.
Would the Sanhedrin/Beis Hamidrash have an obligation to flog or execute him for these crimes? For simplicity, let us ignore any sins committed during gladiator performances.
If not, was it simply because he left fewer than two surviving kosher witnesses?
Was the Sanhedrin's "secular authority" not invoked because he was clearly of no future danger?
In general, acts of theft and violence are exonerated by monetary payments, as Jewish courts practiced restorative justice rather than retributive justice. So there would have been no flogging (malkos), imprisonment, or capital punishment incurred for his crimes. Even monetary fines (e.g. keifel) can only be payed if there are witnesses to the act, as מודה בקנס פטור, so he would be exempt from those as well.
As for repaying the people he robbed, or paying restitution for those he harmed, that is an integral part of the teshuva process. So although the Gemara doesn't mention it, I would assume R' Shimon did make efforts to repay what and whom he could, and applied the rule for גנב ולא יודע את מי גנב for the rest.
Firstly the story took place hundreds of years after Sanhedrin stopped judging capital cases.They stopped doing so even before they disbanded.
Secondly, even had Sanhedrin been around and even assuming Reish Lakish was guilty of murder (do you have a source that he was? the Gemorah does mention him killing people but they were murderers who were going to kill him )he would have needed two kosher witnesses AND a warning a few seconds before he committed the act of murder that he was doing a capital. offense. It's safe to assume he didn't meet that criterion. He would have had to meet the same criteria for lashes. Furthermore, theft is not a crime that one receives lashes for.
One can question if he paid back what he stole. There is no reason to assume he didn't but if he didn't it may inter alia be because of the Gemorah which says that if a thief is looking to do Teshuva we don't accept the return of his stolen items out of concern that he may decide against doing Teshuva due to the difficulty involved. There are also other reasons why he may not have been halachicly obligated to pay back.
As far as why Sanhedrin didn't use their (nonexistent at the time) secular authority, perhaps that question could be asked about before he did Teshuva but afterward? Sanhedrin was not in the business of using extrajudicial powers to kill people who clearly had done Teshuva.
Decades before they disbanded they stopped judging capital cases because there were too many people who halachicly deserved the death penalty. They certainly weren't giving the death penalty to someone who halachicly didn't deserve it and had clearly done Teshuva
Who said Resh Lakish wasn't punished? But he was a great scholar, which puts him in a different category. As he himself said (and it was apparently accepted):
Resh Lakish said: If a Torah scholar sins, he is not ostracized at all in public, as it is stated: “Therefore, shall you fall in the day, and the prophet also shall fall with you in the night” (Hosea 4:5). This is explained to mean: If a prophet or any other important person sins, his offense should be concealed like the night and not punished in public. [Moed Katan 17a]
