Your reasoning is correct, but it seems to me that introducing the changing of $B's$ origin was kind or arbitrary. Usually, when we talk about Lorentz transformations, we do not talk about looking at individual lines from different frames in reference. Instead, a much easier way would be to actually make a change in the whole co-ordinate system itself.
When you are talking about normal situations, when we consider only two co-moving people, the normal <PERSON> transformations hold, without actually needing spacetime diagrams. These are $$t' = \frac{t - vx}{\sqrt{1- v^2}}$$ $$ x' = \frac{x - vt}{\sqrt{1 - v^2}}$$ assuming we choose units with $c = 1$. These are the Lorentz transformations that you reasoned with only $A$ nd $b$ in account.
Now, when we rather talk about more objects, like in the situation where one frame of reference has both $A$ and $C$, we prefer to use spacetime diagrams. But, then it is becomes less convenient to do the <PERSON> transformations on each and every worldline and measure values. So, we do a trick (which I will briefly describe here, and leave the nitty-gritties to you):
1. Take any one object, for example $A$, and look how its worldline transforms in $B's$ frame. In your case, that was the first part of your derivation.
2. Using the results you obtain create a matrix. By now, its seems apparent where this is going.
3. Take the matrix you obtained and then apply it to the entire $A$ frame of reference (the co-ordinate system where $A$ and $C$ are located). This is standard linear algebra.
The new co-ordinate system that you get will contain every worldline ($A$ and $C$ in this case), from the reference frame of $B$. Yes, your reasoning was correct and works, but is not very convenient for frames with many objects.
Update: I have been asked in the comments to touch upon the second update in the question. It is a way of deriving the $\gamma(v)$ function. 499
But here is why that mathematically yields results but is not that correct.
1. Anything involving faster than light travel is generally avoided due to the fact that special relativity tells us that nothing can travel faster than light. So, even in derivations and mathematics, it is preferable to avoid faster than light situation or reference frames.
2. Time is not interchangeable. While spacetime diagrams are a very powerful way to model relativistic situations, they seem to imply that time and space can switch places by rotations. Physically, this is not possible: sure, you can make changes to the time and space axes, but you cannot switch them. (In very high gravity situations, like inside black holes, time and space can switch but then we need to take into account general relativity, which in this context is out of scope.)
So, though you get results, that is not actually the correct way to do it. Instead here are some clues to what you can do.
1. Leverage symmetry: symmetry is a very important concept in physics, and in deriving the $\gamma(v)$ there are two main symmetries which come into play. I won't just straightforwardly reveal them, instead for now, I will just say that they have to do with directions of velocities and the fact that all observers consider themselves at rest.
2. Substitute: When I learned the derivation of $\gamma(v)$, it used the fact that <PERSON> transformations work both ways: $A$ to $B$ or $B$ to $A$. So you substitute the values of one into another and work your way through it.
3. Work your way through it: You inevitable have to do some very length but easy algebra. So instead of trying to find shortcuts, just go through it the ugly way, and the results you get will be simple and elegant.
If these sounded vague, it is because I don't want to give away the derivation steps completely. Rather, it is meant as an exercise to the reader to try and do this task. 562