Does gravitational time dilation affect apparent mass? Suppose I'm on a highly elliptical orbit around some massive body. As I get closer, I start to experience time dilation due to the object's gravitational field - time starts passing slower for me than for an observer farther away. This means that, according to my frame of reference, my orbital period appears to be decreasing, given that none of my other orbital elements would likely appear to change (both to me and to the outside observer). Does this mean that I would measure the mass of the body as greater than the external observer would?
Another, secondary question that might stem from this is whether, at any point, this (or the fact that my engines' exhaust would appear to have a lower velocity to the external observer) would significantly affect the gains from the Oberth Effect, and if so, what the effect would be and at what point it would start to matter.
 A: In the extreme mass ratio limit, orbits are still described by two exact parameters, corresponding to the energy and angular momentum per unit mass of the orbit as observed from infinity.  Even in a highly elliptical, precessing orbit, these parameters do not change.  If the residents of the satellite know general relativity, then these effects will be accounted for automatically.
The details of the derivation are provided in nearly every book on general relativity, but doing it from scratch, starting from just the Schwarzschild metric would take more text than you could properly do in a post here.  
A: 
Another, secondary question that might stem from this is whether, at any point, this (or the fact that my engines' exhaust would appear to have a lower velocity to the external observer) would significantly affect the gains from the Oberth Effect, and if so, what the effect would be and at what point it would start to matter.

When time dilation factor is x%, then the inefficiency of the rocked is decreased by x% by the time dilation. 
Proof:  Photon rocket emits x% less heat per burned kilogram of fuel, when the burning happens in a gravity well that causes x% gravitational redshift of photons.
Just take the viewpoint of a distant observer and use the law of conservation of energy. Same amount of energy that goes into gravity well will come out of the gravity well. Less energetic photons come out of the gravity well, so more energetic rocket comes out of the gravity well.
A: After continuing to research this, it turns out that other aspects of the orbit would, in fact, change; this is evidenced by the precession of Mercury. I was asking about what would happen to the Newtonian version of an orbit when subjected to time dilation, when to get time dilation you need to essentially scrap Newtonian physics altogether. Orbits aren't even really conic sections; they're geodesics. I had a flawed premise.
I'd still be interested to see what time dilation would do to the Oberth effect, however, if someone could help me with the math because, honestly, I'm still a little shaky with the math behind General Relativity.
