I have been pursuing a line of thought that Akavia represented a distinct school from that of Hillel and Shammia and the defining characteristic of his school was extreme individualism. With this in mind, a certain well known Talmudic passage becomes more intelligible.
Our Rabbis taught(Shabbos 31a): A certain gentile once came before Shammai and asked him: "How many Torot do you have?"
"Two," he replied: "the Written Torah and the Oral Torah."
"I believe you with respect to the Written, but not with respect to the Oral Torah. Make me a proselyte on condition that you teach me the Written Torah [only].
Shammai scolded and rejected him in anger. When he went before Hillel, Hillel accepted him as a proselyte. On the first day, Hillel taught him, "Alef, bet, gimmel, dalet." The following day Hillel reversed the letters.
"But yesterday you did not teach them to me like this," he protested.
"Must you then not rely upon me? Then rely upon me with respect to the Oral Torah as well."
There is a Halachic problem with what Hillel did, which Rashi points out. Talmud says in Bechoros 30 that a convert who accepts all of the Torah except for one matter is not allowed to convert. How could Hillel take in a convert who did not accept the Oral Law. Rashi answers that this convert did accept the Oral Torah; he just didn't believe that it was given by Hashem. This is a kind of answer that generates volumes of discussion, and it did exactly that in the subsequent generations.
It appears to me, however, that this convert did believe in Oral Law but he did not trust either Shammai or Hillel to transmit their versions of it to him. This is why he said: "In regard to Oral Law, I don't believe you". He did not say that he did not believe in it, only that the did not believe Shammai and HIllel. He was quite an individualist and he wanted to learn it from someone like Akavia ben Mehallel.
You see from here an indication, if not a proof, that an alternative formulation of Oral Law existed in the days of Hillel and Shammai, presumably that of Akavia ben Mehallel.