To someone traveling near the speed of light, they are traveling at the speed 0 relative to themselves, so they will not perceive any mass increase. The mass increase is only what stationary observers would detect/measure.

BTW, in the 80s I created a 3D relativistic simulator. Unfortunately it requires a DOS operating system to run :-( and I lost the source code so I can't port it. But if you have a DOS computer or can run DOS in a VM, I am happy to share it. The simulator displays clocks on various points of the moving object so you can see how time dilation occurs, and one can also turn on and off the effect of travel time of light. You can see Terrell rotation occur. You can define your own object shapes, and select any vantage point.