Hopefully I have understood your situation correctly, if not then please let me know and i'll delete this answer.
For a radially in-falling observer to hover at $r = 2M + \epsilon$, they would need to provide an opposing acceleration of $a \sim \frac{1}{4M\sqrt{\epsilon}}$, as you stated.
Any amount more than this critical amount will cause the observer to move radially outwards, away from the black hole. If the observer had a rocket strapped to her back, then she would indeed be propelled outwards by the additional acceleration provided by the hawking radiation, and any infalling observer would see nothing special: just a radially boosted observer sailing past.
The rope in your problem makes this tricky however. If the rope is tied to some fixed point far from the black hole, then you can not ride the hawking radiation from the black hole. This is because the moment you move radially outwards from your position, the rope will stop providing any force, as it presumably goes slack. Thus, gravity will immediately grab a hold of you again and drag you back to $r = 2M + \epsilon$.
If your rope is tied to an accelerating rocket or you happen to have luckily grabbed on to the tentacle of a giant galactic space squid desperately trying to save you, then you will again be in the situation where it is as if you have a rocket attached to your back and an in-falling observer will again see nothing weird.
With regards to whether or not you can escape a black hole eventually powered only by Hawking radiation, you might want to take a look at the calculation that has been done in this paper (see also this paper).
Conceptually, a hovering observer will measure hawking radiation and this will give an acceleration to the observer, potentially allowing them to escape the black hole. However, with regards to turning off the acceleration that keeps you static, you would need to wait until the amount of hawking radiation was large enough to sustain your motion without the tension of the rope.
This may in principle be possible, but you would have to wait for an incredibly long time, since black holes evaporate slowly and give out very little hawking radiation until they have very small mass. At this point, quantum gravity becomes important so who knows.
Another point is about the sail. By calculating the acceleration at $r = 2M(1+\epsilon)$, you are only getting information about what the acceleration would be there at some fixed $\sigma$, which would die out as you got further away. In order to calculate this more effectively. To maintain the same acceleration, you would need to increase the area of your sail as you moved out.
The power is given by
$$
P \propto A_{BH}T(r)^4 \propto \frac{1}{M^2(1-\frac{2M}{r})^2}
$$
While the acceleration by
$$
a_{hawking} = \frac{P}{\sigma} = \frac{A_{sail}}{mM^2(1-\frac{2M}{r})^2}
$$
Where $\sigma = m/A_{sail}$.
This needs to be bigger than the gravitational acceleration felt by the observer, meaning our area needs to be:
$$
A_{sail} > \frac{mM^3(1-\frac{2M}{r})^{3/2}}{r^2}
$$
For $r = 2M(1+\epsilon)$, this means that $A_{sail} > mM\epsilon^{3/2}$.
Best Answer
This is not a full answer, since I don't know the full answer, but it is more than a comment.
My contribution is to compare your question with another, simpler one, and then come back to yours.
1. Observer-dependence of radiation in classical physics
Here is the simpler question (one to which an answer is known). A charged particle accelerating in empty space will emit electromagnetic radiation. A charged particle fixed at some point in a static gravitational field will not emit electromagnetic radiation. Both statements are true---relative to a certain natural choice of reference frame in each case. But, in the first example one could step on board a rocket accelerating with the particle, and no electromagnetic waves would be seen in the rocket frame. In the second example, one could go freely falling past the particle, and in this freely falling frame, electromagnetic radiation will be seen. So what is going on here? Does the charged particle emit radiation or doesn't it?
All of this scenario can be treated with special relativity, and it teaches good lessons to prepare us for general relativity. The main lesson here is that the process of emission and subsequent absorption of radiation is not absolute but relative, when accelerating reference frames are considered, but the state changes associated with absorbing radiation, such as a detector clicking, are absolute. It is the way we interpret what caused a detector to click that can change from one frame to another.
To make the above really connect with your question, note that in my simple scenarios I could imagine a cloud of gas possibly absorbing the radiation in between emitter and receiver, just like in your scenario.
2. To resolve a paradox in observer-dependent physics, first convince yourself about the easiest observer, then seek arguments to explain what the other observer finds
The above principle can be applied to resolve puzzles in relativity such as whether a fast pole can fit into a short barn, or whether a fast rivet can squash a bug in a hole.
3. Unruh effect has two complementary physical interpretations, depending on who is accelerating
Hawking radiation is like Unruh radiation, and therefore it is more subtle than classical radiation. A useful tip from the consideration of Unruh radiation is as follows. Unruh's calculation says a detector accelerating through the vacuum picks up internal energy, equivalent to detecting particles. Now if we look at this detector from an inertial frame, we still conclude it picks up excitation, but we interpret differently: we say the force pushing it provided some energy which got converted into internal energy because the process is not perfectly smooth.
4. Answer
Now I will apply all the above to provide an answer to your question. I admit I am not sure and the following answer is just my guess. I only claim it to be an intelligent guess.
My guess is that the distant observer observes Hawking radiation and absorption lines. I say this because it is a consistent summary of what seems to me to be ordinary physics, assuming that there is Hawking radiation coming up from a horizon.
So the puzzle is to explain this from the point of view of the freely falling cloud. I think an observer falling with the cloud looks up at his distant friend and notices that his friend is accelerating through the vacuum, and consequently experiencing internal excitation owing to fluctuation of the forces accelerating him. To account for the absorption lines, i.e. the absence of excitation at certain frequencies, I guess (and this is the speculative part) that now the calculation from quantum field theory would have to take into account that the rest of spacetime is not empty but has the cloud you mentioned, and this cloud affects the overall action of the quantum fields in this way. Note, it is not a case of action at a distance (in either perspective), but it is a highly surprising prediction so I think your question is a very interesting one.