One of the major factors determining the stability of a Trojan object is the mass ration between the planet and the Sun. Since the Sun is some 330,000 times more massive than the Earth, this allows a bit of leeway for the mass of the Trojan object.
To text the idea as to whether Mars would be stable at as an Earth Trojan object, a ran a grav-sim simulation.
It is crazy to me you can sim that information. Love it. Do you have access to predicted bodies coming into contact with Mars in a way that might push it our way? Seriously, I agree the amount of force needed would be to great for humanity to harness in the next few thousand years. My only realistic hope would be a large body to zoom through the solar system and graze Mars. And I do not believe Mars would have to become a Trojan in Earth's orbit.. Once I believe it likely it was Achaean with us but considering the physical nature of Mars currently... I think it would need to orbit slightly closer to the Sun than we are. C/2013 A1 appears to be to far off to have a wake that could move it. Also if the asteroid is moving outward instead of inward that's no help
Since perturbations from other bodies can effect the stability I added a Jupiter to the Simulation. after 15,000 simulator years, Mars still showed no signs of changing its relative position with respect to the Earth.