Your question is throwing some people off, because you used the words point mass and acceleration. Really, you are considering two distinct point masses whose initial states differ only in their velocities. The inertial reference frames attached to the particles are said to be related by a boost. It is practical to talk about abstract inertial observers instead of concrete point particles located at the origin of inertial frames of reference. It simplifies the language and makes it clear we are only interested in the properties of space-time itself.
Your argument is solid. Accepting the contraction of space due to the Lorentz transformation forces us to conclude that spatial ordering is relative. I really like the setup and notation you have used to demonstrate this. The way you apply special relativity is a little sloppy though. I'll get to that later, but as general advice I would advise you to always reason in terms of events. Events are nice, because in the case of multiple observers it is always clear what is being measured: the event. The observers can simply associate a space-time coordinate to the event, which we can then compare. If we ask multiple observers what the distance between two given observers is, it is not clear at all what we're asking of the observers.
Answer
The standard approach to resolving the paradox is to remember that simultaneity is relative, draw a space-time diagram and show that everything is good and well. I'm already boring myself, so let's try something different.
One of the central themes of special relativity is that space and time stand on equal footing. For every spatial concept, there is a corresponding temporal concept. We have space contraction and time dilation, energy and momentum, density and flux and so on. You have created a spatial paradox. What is the corresponding temporal paradox? I'd like to use your notation, but I'm going to refine your argument a little bit.
Consider three inertial observers, $A,B$ and $C$. Observers $B$ and $C$ are at rest with respect to each other and $A$ is not. 562
$A$ observes that $B$ and $C$ are moving towards him and he measures that $B$ passes him (an event) after one unit of time passed on his clock and after having travelled one unit of distance relative to $B$. He passes $C$ after two units of time and two units of distance. In your notation, slightly adapted, the space diagrams are
$$A:A-B-C$$ $$B:A-B\ \ |\ \ C$$ $$C:B\ \ |\ \ A--C$$
The vertical bar represent an infinite amount of dashes. From $B$'s perspective, for example, the distance he travels before he meets $C$ is infinite, because they never actually meet. I am forced to flip $A$ and $B$ in the third diagram, because it is impossible to fit an infinite distance ($B\ |\ C$) within a finite distance ($A--C$). Observers $B$ and $C$ disagree. According to $B$, at the beginning of the scenario $A$ is to the left of $B$. $C$ says $A$ starts off at the right of $B$. This is your spatial paradox, slightly reformulated.
The corresponding time diagrams are exactly the same as the space diagrams. The dashes now represent the time intervals that are recorded by the observers. $B$ and $C$ are once again saying something very different. According to $B$, it will meet $A$ in the future. $C$ says $B$ has already met $A$. This is the temporal paradox.
In summary, the scenario starts out as follows. According to $B$, $A$ is on the left of $B$ and will pass $B$ in the future. According to $C$, $A$ is on the right of $B$ and has passed $B$ in the past. These two statements do not contradict each other at all. We see that the spatial paradox has a temporal twin and that they resolve mutually.
Wouldn't it be neat if other paradoxes in special relativity worked like this? 272