"Even if we once knew Christ according to the flesh,
we no longer know him that way."
Taking this statement in the context of Acts and the rest of Paul's
letters I would suggest:
(1) CRISTOS is here only remotely connected with Jesus' messiahship. As
almost everywhere else in Paul the word is used as a proper name.
(2) It is clear from Paul's letters that he took very little interest in
the words and deeds of the historical Jesus. Therefore the statement
makes sense as Paul not only admitting the fact, but also claiming that
this attitude constitutes a morally superior position.
(3) This interpretation fits perfectly with Paul's known antagonism
towards James the brother of Jesus, and (to a lesser extent) Peter. They
must have helped to propagate the sayings of Jesus, some of which found
their way into Q. Peter may also have told Paul a few stories about
Jesus during their fortnight together in Jerusalem (Gal 1:18). But Paul
was fiercely independent, and didn't want to be reliant on the Jerusalem
"pillars", which he would have had to have been for the stories and
sayings of Jesus.
(4) The element of boasting in verses 16-17 is typically Pauline, and
there is no way the statement is an interpolation.
Weston-on-Trent, Derby, UK