That Oswald did not realize that Red Robin and Robin were the same person was not something that he felt the need to clarify. In his new guise he taken to a harder edge, even before the death of Bruce, so that he found it convenient that his new identity was distanced from that of the iconic Batman and Robin.
Unfortunately much of what he said concerning the GCPD was probably true. There were, to be sure, very good and true cops among their number but the corruption remained. It likely always wound in some form. But that and the jabs from Oswald did not mean anything to him, but the mention of who the killer is, well, that was another matter entirely.
As far as Tim could see, he had two options. In the first, he asked Oswald for the names of the killers. He would refuse, and be beaten further for it. In the second option, he would play ball with Oswald. Either way he did not trust the man as far as he could throw him, but if there was even the slightest chance that he was telling the truth, he had to hear it. With the city about to break apart he needed to find the killer as soon as possible, so that he could focus on other things. Mourning included.
Of everything that Oswald said, Red Robin responded to but one thing. The offer itself.
"If you tell me the names of the killers, I will allow you your retribution. But if on the other hand I find that you have mislead or otherwise lied to me, well, let's just say that you will be wishing that you were dealing with the Batman."