How come the dentist didn't tell his patient about the writing on his teeth, and then ask him if he needed help? Presumably, if God put "help me please" on the man's teeth, then he's the one who needs the help being asked for.
Basically, Rule of the Idiot Plot in the world of the Coen Brothers. The utter Fridge Logic is part of the joke.
Why didn't Larry take another, simple option for the bribe money: act like it was found? He did find it on his desk and did try to return it to Clive who turned it down. The office administrator who gives him his phone messages would have witnessed him chasing after Clive after he leaves Larry's office (to return the money) the first time and the second meeting as well (where he denies that the money belonged to him).
Step 1: Like any found item, one would ask around if anyone had lost it. If he let lots of people know he found it (but not the amount), including the students then he has witnesses that he exercised basic due diligence (attempting to find the owner).
Step 2: After a reasonable amount of time (and before the grades go out), when it is not claimed bring it to the police station.
Larry: "Hello officers, I found this envelope containing quite a bit of money but no one has claimed it. I found it on my desk after speaking to a student, Clive Park. Thinking it belonged to him, I asked him if he had lost it but he said it was not his. I've asked around, but no one has yet to step forward to claim it."
Step 3: Get a copy of the police report or whatever would be filled out and then just wait. If the Parks try to bring legal charges for theft, they have no leg to stand on. He didn't steal it; the police have it as lost property and the Parks more than welcome to go claim it. And they can't sue him for defamation since Larry never said that the money was left as a bribe; it was just found on his desk.
Result: If they claim it from the police station, they then have to explain why Clive was carrying around that large amount of cash in a plain envelope to begin with and why he denied that it was his when Larry tried to return it. If the continue the shtick that Larry is a thief, then they would have to explain why it was that they thought that Larry, of all people, had stolen the money in the first place when he went through all that trouble to find the rightful owner (also defamation goes both ways). Attempting to explain would actually just make things look worse for them. It especially looks like malicious revenge since Clive had failed Larry's class.
This absolutely would've been the best idea. However, the reason Larry didn't do it is pretty simple: he's extremely passive. When you look at the rest of the film, it's clear Larry rarely does anything to help himself. He's forced out of his home, yet doesn't really resist it despite hating the circumstances. Hell, his wife and Sy seem to be talking down to him when they suggest he do it. Similarly, he could quite easily clear up this bribery mess through the aforementioned process, but because his superior interrupted him with news over the denigrating letters, he instantly stopped talking. It's not so much a case of Larry holding the Idiot Ball as much as it is him just being such a pushover that he lets things happen to him and does nothing to really resist them. In essence, it's an inverse of Men Act, Women Are; Larry is defined by his inaction, not by what he actually does.