In his brief time in Washington, Sen. Cory Booker has become famous for obsessively shooting selfies with other lawmakers. But now he’s starting to develop a signature issue that will render him even more irrelevant: animal-rights extremism.

Already named by PETA as the “Hottest Vegetarian Senator,” Booker upped the ante recently by announcing on Twitter that he’s gone full vegan.

In a recent Daily Beast interview with animal-rights activist Vlad Chituc, Booker talked as if it’s all personal, explaining how “being vegan for me is a cleaner way of not participating in practices that don’t align with my values.”

Problem is, he plainly wants to impose those values on the rest of us.

In response to a question about “implementing a strong Animal Rights Bill,” Booker tweeted in September that he was working “with animal-rights groups on some legislation.”

Since then he has promoted all sorts of animal-rights propaganda on social media, including: posting a video in December of a turkey taking a bath, and getting blow-dried, as part of its rescue from the evil of Thanksgiving dinner; a message about how you can’t love animals if you eat them; a tweet thanking PETA for its “activism on animal rights,” and an Instagram post implying that you support animal cruelty if you don’t buy vegan shoes.

And his legislative track record shows it’s not just talk. The Humane Society Legislative Fund (the nonprofit’s political arm) praises him effusively for three votes last year, including against the Bipartisan Sportsmen’s Act.

He was one of just 12 senators to oppose even taking up that bill, whose stated goals were “to protect and enhance opportunities for recreational hunting, fishing, and shooting.” (The bill later died in a fight over gun-control amendments.)

Most recently, Booker joined Sen. Dianne Feinstein in co-sponsoring a bill to limit antibiotics in livestock. According to GovTrack.us, the bill only has a 1 percent chance of passing.

He also tweeted and Facebooked in support of New Jersey legislation to “ban gestation crates for pigs.” Gov. Chris Christie ultimately vetoed that bill, maintaining it was “passed in response to ‘misguided partisans and special interest groups.’ ”

Give Booker credit for living his values and raising awareness on preventing animal cruelty. While Newark mayor, he personally rescued two dogs — one from the freezing cold, and the other after it was confined for four days in a crate.

He also rescued a starving young pit bull named Patrick, which won him an award from The Humane Society (whose legislative arm later endorsed him for Senate).

But he couldn’t restrict it to his private life. He used the Patrick incident to push for a no-kill animal shelter in Newark.

This was just silly: The need to “put down” sick, aggressive, injured or suffering animals may be a sad reality, but it’s one that even The Humane Society and PETA recognize.

At the time, the general manager of the Associated Humane Societies’ shelter in Newark criticized Booker’s no-kill shelter proposal, explaining how it can be more humane to euthanize animals than to warehouse them in cages while wasting money on expensive veterinary procedures.

It’s hard to say how much of Booker’s time went to this absurd crusade — but it’s plain that the mayor of one of New Jersey’s more impoverished cities had better things to be doing for his constituents.

And that’s the biggest problem with his extreme animal-rights agenda in Washington: It’s going nowhere, and New Jerseyans surely have more important work for him to do.

Of course, it does have rewards for him: For the holidays, PETA named a rescued chicken in his honor and gave him a painting “created” by a chicken with wet paint walking on a piece of paper.