Being woke is a political tool and is used as a rhetorical club to denigrate others, not help anyone. Jesus is apolitical, and not woke. Today, He would not endorse any candidate or platform, and if He met with world leaders He would call them out on their sin, not let them lie to themselves or to us. "You'll save $2,500 a year and can keep your doctor", for example, would not be acceptable. The next time He gets involved in politics, it will be to totally take control, so good luck trying to insist He has to fit in your incredibly small mold.
To you, caring about social injustice and discrimination means agreeing with a political party so you can feel self-righteous and criticize others for disagreeing. To Jesus, caring about social injustice and discrimination means loving people enough to accept them where they are while calling them to repent of their sin and selfishness. It means getting involved in people's lives to the point they trust you to help them, NOT expecting a government program to take over and throw confiscated taxpayer money at them. You know, things that the "woke" hate, because they don't require government control and are voluntary in nature. It means hiring people to do work even if they posted on social media that they are sympathetic to people getting lengthy prison sentences for merely walking into the Capitol building and leaving. It means having the courage to gently and lovingly let people know that they really are not a cat just because they want to be one and they cannot take a dump in the lobby while working for you. It means being willing to sacrifice our own comfort to let others know they're really cared about, not just give lip service to a government program.
Being woke, using today's definition, involves not only allowing people to lie to themselves and others but validating those lies. Therefore, Jesus is not woke, because He is all about truth, not lies. Now, you're trying to be cute, but as I've said before, Jesus doesn't care about your politics, He's far bigger than any political party or platform, and He cannot be shoehorned into any political ideology. Those who attempt to use Him for political purposes are simply wrong.
One final thought, if you are indeed trying to bring Jesus in to bolster your argument for a political ideology, are you not attempting to set up a theocracy? Or are you planning to simply jettison Him as soon as you achieve the power over other people's lives you're trying to gain?