I finished modeling the new Y-axis parts to add a 3mm linear rail for stability.

I modified the Y-axis slide slightly from the picture above so it could be printed without supports, and also to solve a potential interference.

I printed the new parts and assembled it, two minor issues found, first inserting the LM3UU bearing caused the slide to bend slightly, and second the Y-axis motor was slightly too far away, so it can't be screwed down completely without binding. Here's a photo of the new parts assembled, unfortunately I didn't have time to test it out, but it seems the backlash is slightly improved, but still present due to the tolerances of the LM3UU and the X-axis slide (which might be slightly improved by adding an LM3UU to as well, but it is probably not worth the effort):

Another thing to note, the entire X-Y assembly is a bit too tall and interferes with the 100x objective, so I'll probably 3D print an adapter to mount it next to the focus plate instead of on top of it.