Looks like there is no way to export Maya's trax clips. Maya's DAE exporter only outputs the main timeline as a single animation. On the other hand, FBX exporter lets you define parts of the main timeline as named clips. The clips and their tracks are properly recognized by Godot but some track's keys are null-ed. Couldn't get it to work properly.
I ended up using DAE with all animations on main timeline, slicing it into clips in Godot's import options. Not ideal but it works. Will probably switch completely to Blender for more complex stuff.