1. The purpose of the visit to Jerusalem of Gal 2:1-10 was to discuss the circumcision question. This is clear from Acts 15:1-2. In Gal 2:2 suggests that Paul had received a revelation to the effect that he should go up to Jerusalem to lay out the gospel that he preached among the Gentiles (i.e. his gospel of non-circumcision). Further, Carlson's work suggests that the men from James of Gal 2:12 had arrived in Antioch before Paul's Jerusalem visit and could have precipitated it (see my discussion here).
Now, if Titus, like Timothy, had Jewish heritage, the circumcision question would have been particularly relevant for him, and this could explain why he was selected to accompany Paul and Barnabas to Jerusalem.
2. Titus's Jewish heritage, if known to the Galatians, would also explain the "not even" (οὐδὲ) in Gal 2:3. We could then paraphrase: "Not even Titus (whom you know had Jewish heritage), ... was compelled to be circumcised.
3. Back in 1998 Christopher Hutson sent me useful feedback of an early form the Titus-Timothy hypothesis, and included the following paragraph, which I reproduce with permission.
"On the other hand, it seems to me that Gal 2:3 is potentially very important for your case. As I rad this, it strikes me that the modern critical editions have an odd punctuation here, setting off the words hELLHN WN with commas, as if they were a separate phrase. But to my eye, it makes much more sense to treat the participle WN with the definite article hO, so that the entire phrase hO SUN EMOI hELLHN WN is one grammatical unit, not two. This seems to be the way Lightfoot read it. So Check WH and other critical editions prior to NH26 and UBS3. Notice also that EMOI is emphatic (as opposed to the enclitic MOI). So you could translate, "who was with ME as a Greek." For your purposes, you might consider whether this implies that Titus presented himself in other circumstances as a Jew. That would suggest that Titus, like Timothy of Acts 16, stood between Jewish and Gentile worlds. Hmmm."Hutson now prefers to translate, "who with ME was a Greek", which is nearly the same. Now, I don't think we can prove that EMOI is emphatic, but it is an attractive possibility. Hutson's reading does have the clear advantage that it avoids the cumbersome commas, and I think Askwith suggested a similar reading. It raises the possibility that in Gal 2:3 Paul alludes to an occasion, known to the Galatians, where Titus had passed as a Jew. This would create a rather smooth transition to the next verses, where this occasion seems to be mentioned:
4. Gal 2:4-5 is explicable if the "false brothers", through their spying, had found out that Titus was not actually a Jew. These verses, therefore, contain a further hint that Titus had, on another occasion, passed himself off as a Jew.
Therefore, we have hints that Titus had some Jewish heritage, was able to pass as a Jew, and had done so on an occasion known to the Galatians, until he had been found out.
We have the same information about Timothy, who had a Jewish mother.
The Babylonian Talmud reads:
‘And Rav also ruled that the child is fit, for once a man appeared before Rav and asked him, "What [is the legal position of the child] where an idolator or a slave cohabited with the daughter of an Israelite?" "The child is fit," the Master replied. ..... Rav Yehudah also ruled that the child is fit, for when one came before Rav Yehudah, the latter told him, "Go and conceal your identity or marry one of your own kind." When such a man appeared before Rava he told him, "Either go abroad or marry one of your own kind."’ (b. Yev 45a-b)
This passage concerns advice for men who, like Timothy, are born to a Jewish mother, and a gentile father. Yehudah suggests that they conceal their half-gentile parentage. Rava gives essentially the same advice, suggesting that they go abroad (where their gentile fathers will not be known). Thus, we should not be surprised if Timothy passed as a Jew on some occasion. Acts 16:1-3 says that Timothy was circumcised because it was known (by then) that his father had been a Greek, suggesting that he would have passed (or continue to pass) as a Jew if his Gentile status had not been known.
Putting the pieces together we then get the following reconstruction: Titus-Timothy was born in Antioch to a Jewish mother and a Gentile father. He went to Jerusalem with Paul to help resolve the circumcision question. Paul sent him to (south) Galatia to organize a collection. He passed himself off as a Jew there, until some 'false brothers' found out that he was a Gentile. He then had to be circumcised (Acts 16:1-3). Paul later wrote to the (south) Galatians: "Not even Titus-Timothy [the half-Jew], who with me was a Greek, was compelled to be circumcised. But because of false brothers secretly brought in, who slipped in to spy on the freedom we have in Christ Jesus, so that they might enslave us - we did not submit to them for an hour so that the truth of the gospel might always remain with you [Galatians]."
I'm not saying that Gal 2:1-5 demands to be read in this way, but it surely does work well as commentary on the events of Acts 16:1-3.