Every experiment that we've ever performed is consistent with isotropic propagation of light, and yours would be too. To the extent that science can ever know anything, we do know that the propagation of light is isotropic, and your experiment would confirm it.
This is about the fifth question about the new Veritasium video that I've seen on this site today, so I finally watched it. It's pretty cranky. That the one-way speed of light has never been measured is in fact a common claim of aether-theory cranks, and the argument in the Veritasium video is the same that they use.
The argument amounts to the following: given inertial coordinates $(x,t)$ with respect to which the speed of light is constant ($|dx/dt| = c$), you can always define, say, $t'=t-x$, and with respect to those coordinates the speed of light $|dx/dt'|$ ranges from $c/2$ to $\infty$ depending on direction. That's the entire content of the video's claim that the light from Mars might "really" be traveling infinitely fast to Earth. It's purely a formal substitution of variables and there's no physical meaning to the different speeds.
The description of your experiment in nonisotropic coordinates depends on the particular coordinates that you choose. With respect to the $(x,t')$ coordinates above, the explanation is that the angle from the two lasers to the viewer is different, and so the speed of light along the two paths is different, and this together with the speed of the light between the lasers gives you exactly the same delay you'd get in the inertial coordinates. You could specify the experiment more precisely and go into more detail but it's really just a matter of substituting $t'+x$ for $t$.
The sensible way to look at it is the other way around: the fact that there exist any coordinate systems with respect to which the speed of light is constant is what we mean when we say the speed of light is constant. In principle we could live in a world in which there are no such coordinate systems (even locally), and in that world we wouldn't say that light speed is constant. Likewise, rotational symmetry means there exists a transformation of all of the fields that preserves the physics under rotation. It would make no sense to say that because there are also transformations that don't work, that we don't really know that the world is rotationally symmetric. That's what Veritasium is saying.