In other words, when a particle's mass increases significantly as it approaches the speed of light (c), does its influence on gravity increase, and if so, does it increase proportionally?
The mass (the true mass) is an intrinsic property of a body, i.e, it does not depends on the observer's frame of reference. The mass used when calculating the gravitational force, via Newton's law, does not change. So in answer to your question: "No, the gravitational force doesn't change in that situation".
I strongly suggest to read this popular article by Lev Okun where he calls the "relativistic mass" a pedagogical virus.
The concept of a so-called relativistic mass which changes with velocity is no longer considered the best interpretation of mass in special relativity. Instead, the invariant/rest mass is used. See here for a discussion. The increase in energy (through momentum) of energy associated with speeding up, however, does increase the extent to which an object contributes to the stress-energy tensor of the system under investigation. Hence, it is a stronger source of gravity.