Can a Catholic participate in an effort that furthers the political ends of a politician who holds views contrary to the Catholic faith?

The Catholic Faith requires a member of the Church be guided by Church teachings regarding moral issues in casting their vote. That doesn't mean that you can only vote for a candidate whose stances on issues are 100% in line with Catholic teachings.

There is a concept in Catholic moral reasoning, which is based on the Natural Law, called the principle of double effect. There are numerous articles on this you could read by searching online. I'd recommend anything from Catholic Answers (catholic.com). This principle states that, as long as an act is not intrinsically evil (ie it is not evil in and of itself), it can be taken with the goal of bringing about a good even if there are unintended yet foreseeable negative consequences. The only other caveat is that the good must be greater than the evil. So, for instance, if a person could save more lives by directing a train to go down one set off tracks instead of another, even if some lives will be lost by the train going down the new tracks, it is morally permissible to switch the train. In the opposite case, where more would be injured by taking action, that action is not permitted because of the second requirement (greater/equal good must be brought about than evil). In a similar scenario, say you could save hundreds of lives by murdering just one innocent person. You can't do that under double effect because the act of murdering, of killing an innocent person, is always evil.

Now that we've explained double effect, we need to apply it to voting. A Catholic must consider all the foreseeable outcomes of casting a vote. If a candidate says they will stand for life and, once elected, expands abortion, that is not foreseeable. The candidate just straight up lied. But if we assume that candidates are likely to do the things the campaign on and act in accordance with prior political behavior (assuming they have a prior political career), we can predict what sorts of things they will do if elected or re-elected. We also have our individual nations' bishops' councils to guide us. In America, for instance, our bishops have told us that abortion is the pre-eminent moral issue for our nation. That might not be true in other nations. For instance, in North Korea, the fact that people are constantly being harangued by a maniacal tyrant might take precedence because of its severity. But when an American Catholic votes, he/she is obligated to follow the bishop's advice in casting their vote. So, abortion should be the weightiest issue when voting.

That doesn't mean that a Catholic cannot vote for a pro choice candidate, but the opposing options would need to be far, far worse. If Biden were running against Adolf Hitler, I'd vote for Biden. When he's running against a milquetoast leader like Trump, who has a proven track record of helping the pro life cause, Trump is far more likely to come out ahead. This is because of the weightiness my bishops say must be put on the issue of abortion.

Other issues come into play, too. Perhaps if there were a candidate who stood for all things Catholic save abortion, and another candidate who stood for all things anti-Catholic save abortion, one might feel the need to vote for the former candidate, and that is reasonable in such a scenario.

There is, furthermore, nothing wrong with advocating for a political candidate, and thus the same rules as voting would seem to apply to advocacy.

All this to say that, essentially, the answer to your question is Catholics will often need to advocate for the "least bad" candidate, at least in the modern Western world, and that that means furthering the political ends of a candidate whose views partially diverge from Catholicism.