Mass is an invariant. Its not "relativity" that has ever held otherwise. It is an increadibly outdated definition made prior to our improved understanding of relativity in terms of the full tensor modeling of the laws of physics that you are stuck on. With the understanding of relativity that we have now mass in terms of the force law you are missunderstanding is the proportionality constant between four-vector force F and four-vector acceleration A in
F = mA
It does not change with speed which has to be the case in the full tensor equation because the very postulate of relativity is that the laws of physics do not depend on frame. The mistake others have made and you are following is that when you transform the proper time derivatives in the definitions of F and A to the labs coordinate time due to time dilation the expressions you end up with have factors of γ that you then missassociate with the mass even though they actually had nothing to due with it. They were due to time dilation. So called mass dilation experiments that you are referring to are not actually confirming mass dilation, but are actually confirming time dilation and the tensor law.