Could a Christian remain in the sect of the Pharisees? Acts 15:5 reads:


  But some from the religious party of the Pharisees who had believed
  stood up and said, “It is necessary to circumcise the Gentiles and to
  order them to observe the law of Moses.” (Acts 15:5 NET)


This verse does not explicitly state that these Pharisees left or were expelled from the sect upon converting to Christianity.

Are there any historical documents that discuss what would have happened to a member of the Pharisees upon conversion to Christianity?
What biblical evidence there is, suggests that there may have been a small window in history (~30-35 years) when this was indeed possible.  As well as the verse you've already cited, the Apostle Paul appears not to have been formally excluded from the sect prior to his appearance before the Sanhedrin in Jerusalem detailed in Acts 23 (~57AD). This is seen by virtue of the fact that his appeal to his fellow Pharisees was not dismissed out of hand, but divided the council:

6 Then Paul, knowing that some of them were Sadducees and the others Pharisees, called out in the Sanhedrin, “My brothers, I am a Pharisee, descended from Pharisees. I stand on trial because of the hope of the resurrection of the dead.” 7 When he said this, a dispute broke out between the Pharisees and the Sadducees, and the assembly was divided. 8 (The Sadducees say that there is no resurrection, and that there are neither angels nor spirits, but the Pharisees believe all these things.)
9 There was a great uproar, and some of the teachers of the law who were Pharisees stood up and argued vigorously. “We find nothing wrong with this man,” they said. “What if a spirit or an angel has spoken to him?”

If we look beyond scripture to the writings of Hegesippus (as relayed by Eusebius), it seems likely that this window closed about the time of the death of James the Just in about 62AD (source).