There is a third possibility: He wasn't an ordained Rabbi.
He was called rabbi out of sense of respect and he was allowed in the synagogues because of his fame. When he preached at a synagogue it says -
Luke 4:20 "The eyes of everyone in the synagogue were fastened on him."
This suggests to me an event like that of a visiting celebrity than a routine occurrence.
Another important thing is the meaning of the word Rabbi. We must ask what the word meant to those who used it, not what it means to us today.
Matthew 23:8 "Don't let anyone call you 'Rabbi,' for you have only one teacher."
If the people used the word as a title of honor or synonym for teacher that wouldn't make him a Rabbi in a literal sense of the word.
Besides, it seems unlikely to me that Jesus would have joined a synagogue and got himself ordained.
He spent a great deal of time criticizing the religious system of that day including the hypocrisy of the Rabbis. He couldn't have stayed quiet in the face of any kind of hypocrisy or superstition. It would be out of character for him. I doubt that with that kind of attitude and vocal criticism of other Rabbis he would've been allowed into any synagogue, much less officially ordained.
Jesus couldn't have been a Rabbi, when he so vehemently criticized them. So much so that he got himself crucified for it.