But from my understanding, this change in mass would not be felt from the frame of reference of the moving object, so from that FOR there is nothing capping the speed.
At the center of mass of the system a uniquely, not velocity dependent mass, is called the invariant mass of the system and is the "length" of the four momentum vector describing the system in any Lorentz covariant frame. There is no flaw to your argument.
The limit of the speed of light to c comes from Lorentz transformations .
One has to have a clear understanding of what physics and its theories are. Physics uses mathematics as tools in order to describe observations and measurements, and be able to predict for new boundary conditions. After the time of Newton and the extensive use of differential and integral equations, it became necessary to impose physics "axioms" on the set of possible mathematical solutions, in order to pick up the subset of those solutions that were descriptive and predictive of physical observables. These physics axioms are called laws, or postulates, or principles, as in Newton's laws, or the postulates of quantum mechanics. and the "heisenberg uncertainty principle".
The limit of the speed of light originally came up when Maxwell combined the electric and magnetic laws of the time into an elegant electromagnetic theory, which predicted the existence of electromagnetic waves that transferred energy with a specific velocity in a specific medium. It was Lorentz who noticed that under Lorentz transformations the systems remained covariant, and that is why the transformations have his name, though they are inherent in the Maxwell system.
In classical physics, light is described as a type of electromagnetic wave. The classical behaviour of the electromagnetic field is described by Maxwell's equations, which predict that the speed c with which electromagnetic waves (such as light) propagate through the vacuum is related to the electric constant ε0 and the magnetic constant μ0 by the equation
So, in a mathematically convoluted way the limit of c for electromagnetic radiation is based on the laws of Ampere and Faraday ... , because Maxwell's equations describe the observations based on these fundamental laws.
When one hits on fundamental laws, the answer to "why" is "because it describes and predicts observations and measurements perfectly up to now" .
c became a universal speed limit both in classical and quantum mechanics because of the thinking out of the box of Einstein. trying to reconcile moving frame transformations and Maxwell's elegant and successful theory. The link give an idea of the thought processes.
Thus the general speed limit of c comes as an axiom in the theory of special relativity, which has been validated innumerable times in particle experiments, (for example).
The only answer then to the "why nothing is moving faster than c" is "because the axiom iss necessary to describe all existing data and the theory based on it predicts future measurements perfectly".