uh, no, those aren’t related. inverting the matrix for (cam<-base) gives you a matrix for (base<-cam). neither of these involves the object. it’s not even there conceptually, just the robot/base and the camera.
calculating (base<-obj) requires going “around the corner” that is the camera. the camera is the common frame by virtue of being able to calibrate object and base poses. you have transformations between either leg and the camera. to go between legs (object <> base), you’ll compose the individual transformations. it’s somewhat like adding and subtracting vectors on a triangle, except multiplying and inverting matrices.
(base<-obj) = (base<-cam) * (cam<-obj)
you initially have (cam<-base) and get (base<-cam) by inverting it.
here’s a crude sketch:
I’m not sure what the robot wants. the robot probably wants the target-to-base matrix as is, because that describes where the target is relative to the robot’s base frame. i gather that from what has worked for you so far.
if you invert the target-to-base matrix, you get the base-to-target matrix, i.e. something that describes the base’s origin relative to the object/target. that should be a somewhat useless matrix I think, at least to the robot.