could the Higgs boson field be responsible for the dramatic mass increase incurred when an object nears the speed of light?
No, it could not.
For one thing, it's not really an object's mass that increases as it approaches the speed of light, just its energy. This extra energy, namely kinetic energy, has nothing to do with the Higgs mechanism. The Higgs field itself is only responsible for the masses of fundamental particles - in other words, roughly speaking, it gives fundamental particles a certain amount of intrinsic energy which is separate from any kinetic energy they may have.