For the petite tournedos: Season the tournedos well on all sides with salt and pepper. Heat a small saute pan over medium-high heat. Add the vegetable oil and butter. Add the tournedos and cook for 2 minutes on each side for medium-rare. Transfer to a platter, cover and keep warm.
For the beurre rouge with crabmeat: Add the shallots and garlic to the same pan and cook over medium heat until translucent, 2 minutes. Add the wine, bring to a boil, then immediately reduce the heat to a simmer; cook until the liquid is reduced to 2 to 3 tablespoons, 5 to 6 minutes. Season with 1/4 teaspoon salt and 1/8 teaspoon pepper. Reduce the heat to low and slowly whisk in the butter, a few pieces at a time, until fully incorporated. Add the crabmeat and cook until just warm, about 1 minute.
For the braised baby kale: Heat the oil in a separate medium saucepan over medium heat. Add the onion and cook until just tender, 1 to 2 minutes. Add the kale along with 2 tablespoons water and cook, stirring, until completely wilted, about 2 minutes. Season with salt and pepper. Transfer to a paper towel-lined plate.
To serve: Divide the kale among four plates and top each serving with two beef tournedos. Spoon the crabmeat on top of the tournedos, then spoon the sauce over the crabmeat. Serve immediately.