This point was the lynch pin of W.E. Filmer's argument and David Beyer's presentation of evidence at SBL in 1995 seems to be what persuaded Finegan. However, Beyer's knowledge of contextual Roman events is severely lacking. (For example, he cites Syme that there was no war from 7 BC until 2 BC, but Syme himself dated the Homanadensian War to 4/3 BC.) The bottom line is this: if "twenty-second" is indeed what Josephus originally wrote, it doesn't mean his number is accurate. It would only mean that he contradicts himself... more than once.
First, Beyer did not mention Gaius. At the midsummer hearing in Rome, in the year Herod died, Josephus explicitly places young Gaius Caesar (grandson of Augustus) prominently in attendance. (Antiquities 17.229) But if the teenage Caesar was truly present, this meeting to read Herod's will could not have taken place in 1 BC. Dio Cassius, Suetonius and Velleius combine to assure us that Gaius received the Tribunican power in 1 BC and departed for Syria shortly after his mother Julia was exiled (2/1 BC). Gaius took a leisurely tour around the Agean, finding his way to Samos by late summer of 1 BC, where he saw Tiberius 'the exile'. (Cf. Swan's commentary on Dio 55.10ff.)
If Beyer is right, Gaius' itnerary does not fit, unless Josephus is wrong. So, pick your error.
Second, Beyer did not mention Julias. Josephus also declares that Philip renamed Bethsaida in honor of Julia, Augustus' daughter. (Antiquities 18.28) But if Philip had not become Tetrarch until 1 BC, he would absolutely have known better. As just mentioned, Julia was disgraced in 2 BC, which explains why the designation didn't last very long. It's a wonder Josephus even mentions it, but it only makes sense in 3 or 2 BC.
Again, if "twenty-second year of Tiberius" is true, then Josephus' report about Julias must absolutely be false. Error pickers are now weighing two against one.
There are other examples but these successfully illustrate the point. Josephus may indeed have said that Philip died in Tiberius' twenty-second year. If so, he simply must have been wrong - either about that or about a host of other facts. Since Josephus is elsewhere known to be a bit off in his counting of reigns - an unfortunate trend, we must sadly admit - we have solid precedent for doubting the earlier manuscripts. Even if they are authentic, that number cannot be accurate.
Josephus has to be wrong in at least one of these places. The orthodox solution is also the most economical. Herod died in 4 BC. The occasional and Quixotic attempts to dispute this (as recently as this year, in Novum Testamentum!) really need to be put out of our misery.