Hard to say for sure, the Fiios have been measured by a third party so there's more information on them. The E7+E9 has a DAC, the m-stage doesn't, so if you need one (and it seems like you do) you'll be going over budget with the m-stage. You could always pick up something cheap like the Behringer UCA202 for a DAC, which performs very well for $30.
The E9 has a higher output impedance, not a good thing but won't change the headphones much. Just taking a guess based on the quoted specs for the m-stage, distortion levels will be similar and maybe slightly worse for the Fiios (still very quiet or inaudible). Power output with the E9 is going to be about the same at 60 ohms, maybe a little less at 300 ohms but still more than enough for every headphone with that impedance. The E9 is a slightly noisy amp, but that shouldn't be a concern with the K701. Just don't plug any IEMs into it, the output impedance (which is even higher on the small jack) could play havoc with their frequency response if they're balanced armature IEMs, and the noise may be clearly audible.
The Fiios have an advantage in functionality as well, because the E7 can break off and act as a pretty potent portable amp. It may not be enough to drive the K701 to ear-splitting levels, but it should work perfectly well at regular listening volumes.
