I have searched the internet and have not found any actual test results.
there will be a infinitesimal increase in the mass given by the relativistic expression of angular velocity and inertia, but it will be completely undetectable at the velocities you can rotate a material object without it being teared apart by centrifugal stresses
The relativistic kinetic energy is roughly (forgive me if i'm missing some constant term of order unity)
$$ E = \gamma^2 I \omega^2 $$
the relativistic centrifugal force felt by the material of density $\rho$ is
$$ F = \gamma^2 \rho \omega^2 r $$
strongest materials will stand up to $10^9$ Pa, so for a centrifuge of 1 meter the breakage tangential velocity $ v = \omega r$ will be around $10^2$ meters per second, too far to observe any relativistic increase to the mass. As Forward stated in his 1962 paper, any such experiment at human engineering scales will require some external field to preserve the ring integrity
Other than what is proposed on that paper by Late Dr. Forward, any other gravitational effect for rotating systems is just speculation or pseudo-science. There was some fuzz about this from some guy called Tajmar, but no one ever reproduced his results since 2006, so they are very probably bogus
I have read through the other questions that you have posted on this site and it seems that what you are really struggling with is the nature of gravity. I applaud you for seeking out answers to your questions.
In response to some of the statements you made as an answer to this question, no, a non-rigid object does not behave differently with respect to gravity and rotational forces provided one is considering the entire object. When dealing with situations such as the saucer shedding water, it is vitally important to define questions very thoroughly. Yes, the saucer will weigh less without the water, but the combination of saucer and water (even if that water is on the floor) will have exactly the same mass throughout.
The question becomes one of defining the object in question. Physics questions about situations and objects that are changing are typically framed in terms of "control volumes" or "control masses". If you define your non-rigid object in terms of a "control mass," it will always be the same mass by definition. If you define your non-rigid object in terms of a "control volume," you may find that the mass changes, but you should be able to identify how that happened (like with the water shedding from the saucer).
Ultimately, this question, and the others you have posed about the change of mass of a spinning object, seem appealing because rotation introduces what are called "fictional forces." To an observant layperson like yourself, it seems like a reasonable position that these forces could somehow be redirected to oppose the force of gravity. Unfortunately, intuition is no substitute for hard science. The physics of Aristotle seems intuitive, but when compared to careful observation, they fail.
If you have the patience, and mathematical ability, you might enjoy working through some college level physics books. I think you would particularly benefit from sections on conservation laws. Good luck!