I would sort of like to help out, but I don't quite understand what is at issue. You have two rest frames A and B, and there are clocks at rest in each frame. So what exactly is the issue or question?I'm sorry but we seem to be missing something here - Einstein's first postulate.
Let us look at this in its simplest view.
We have two frames, two inertial frames, in each of which there is a stationary clock. To an observer in frame A clock A is at rest and clock B, at rest in frame B , moving at v relative to A.
Two identical clocks in inertial frames.
Let us suppose they are light clocks.
When the light in clock A has travelled 1 light second to its mirror, it will measure 1 second to have passed and clock A will display (read) 1 second.
When the light in clock B has travelled 1 light second to its mirror, it will measure 1 second to have passed and will display (read) 1 second.
Frame A and Frame B are inertial frames and all we are measuring is how long it takes for light to travel 1 light second in each frame. the same time - 1 second measured from O when the clocks were co-located.
I am not saying that this is simultaneous because that judgement depends on the convention for simultaneity chosen; but I will say that they both happen at the end of equal intervals measured from event O.
Both clocks will read the same at turnround.
However, I am not saying that the travelling clock is not slow - it certainly is - but compared to what? Surely oit must be compared to A's measurement of the turnround time of T' that is the turnround time of B, in A's frame in which the light in clock B has travelled 1 light second to its mirror AND the distance that clock has travelled from clock A.
That is it will have travelledthe distance OT' or γT.