What is the policy for Latter-Day Saints' leaders (like bishops) being politically engaged?

Yes, there are policies about this.

The General Handbook issued to church members and leaders has direct language on the subject in section 20, section 35, and section 38, with 38 being the most relevant for this question.

To summarize them, church property can be used for voter registration or for polling, under certain conditions. Church property cannot be used for fundraising, campaigning, endorsing, rallies, or any other political process. Church records, such as lists of addresses, cannot be used for political purposes. The church does not endorse political parties, candidates, or platforms.

The church does sometimes back specific legislation, though this can only be done under the direction of the First Presidency.

No leader at any level should be giving political statements during church meetings or activities. In this case, "political statements" needs to be defined. The church has a number of issues it considers moral in nature, which it has official stances on and which leaders can discuss, and some of these can be considered political. Abortion is one such topic, which may be considered a political issue but which the church considers a moral issue. Even here, however, leaders should not be discussing which candidates are for or against abortion, what parties or platforms think about abortion, how members should vote in regards to abortion, etc.

Leaders can, and are encouraged, to be politically active outside of their church duties. This means they absolutely can make political statements anytime and anywhere that is not an official church event (ie sponsored activities and meetings), including in the homes of members, in a park, in a store, in a PTA meeting, etc.

Under these conditions, the handbook specifies they should not use their status as an ecclesiastical leader to influence how people vote, and they should not ever claim their opinions are representative of the church or its other leaders.

Interestingly, missionaries are told not to discuss politics at all, using very different language than what is used to instruct bishops and other members.

As has been noted in other answers and comments, bishops are not perfect. I've personally heard bishops make political statements in meetings, though they usually apologize and retract the statement as inappropriate. I understand that even if an apology is given, one can still be influenced by such statements.

Covid and responses to it have become highly politicized and sensitive issues. It is just one example where it can be hard to say what is "political" or "not political", even if no political party/candidate/platform is mentioned. Again, bishops aren't perfect, and some of their opinions will leak out from time to time; such should be addressed with the bishop even if the statement in question is "merely" offensive and not political at all.

Without context for the event in question, I cannot say what you should or should not do (or have done). If the bishop is going against church policies, it should be addressed as any other mistake should be: calmly, directly, and without guile. If the bishop continues to go against church policies, it should absolutely be brought up with the stake president, who will counsel with the bishop to stop his behavior, or possibly explain why the bishop's actions were not in disagreement with church policy. Should a stake president be at fault, there are other authorities to go to.

I decided to change my comment into an answer.

Please be forgiving when I suggest that you need a thicker skin. The Church was much more politically active before 1980. Around that time, the IRS began cracking the whip about 501(c)3 non-profit organizations dabbling in politics and the Church changed its general policies to reflect that. In our modern "politically correct" world we take very quick offense when things aren't "just right," but Bishops (all 31,000 of them) are imperfect too.

As remarked in Ella's answer and your comments, Bishops live in two worlds: one is personal, the other is (ecumenically) "professional." Anyone (like myself) who's had to walk that tightrope knows it's impossible to walk it perfectly.

Joseph Smith once remarked that a prophet is only a prophet when he's acting as such. Bishops, like prophets, have personal lives and are allowed complete freedom within gospel law to live those lives. In other words — your bishop has the right to participate in politics as he sees fit and to use his Constitutional right to free speech. In other words, no bishop has the right to endorse any politician or political issue as a representative of The Church of Jesus Christ of Latter-day Saints, but every bishop has the right to hold his own opinions and to express them.

You've given us no details. You should have. What public forum? Over the pulpit? Talking to a dozen members in front of a store? Chatting with friends at a ward picnic? What did he say? Would you feel the same way if he expressed an opinion that you agreed with? Do you feel that he was using his calling as bishop to unduly influence others toward his opinion?

My worry, since we have no details, is that it's possible the problem is yours, not his. It's worth remembering that not speaking evil against leadership basically means to cut them some slack. I've witnessed prophets do things that raised my eyebrow... but they're human, just like the rest of us, and when I weighed their actions against all of scripture and the Grand Scheme of Things, I found that they were doing the best they could, just like me, and since no harm was done, it was me who had to be more forgiving and understanding.

So, instead of informing on him, try talking to him first.

Per the LDS website, “stake presidents and bishops are free to contribute, serve on campaign committees and otherwise support candidates of their choice”. So, it seems that bishops are free to participate in politics. However, the church goes on to say that these officials should not imply or infer that their political stance is endorsed by the church, use church generated addresses, email systems, buildings etc for political promotions, or “engage in fundraising or other types of campaigning focused on fellow Church members under their ecclesiastical supervision."

From what I can gather, the only members for whom it is not permissible to participate in politics are full time General Authorities, general officers, General Auxiliary Presidencies, mission presidents, temple presidents, and their spouses. The hyperlinked article, which cites a presidency letter the church released that outlined their stance on political neutrality, states that these full time servants “should not personally participate in political campaigns, including promoting candidates, fundraising, speaking in behalf of or otherwise endorsing candidates, and making financial contributions.”