In the above approach is the input signal still considered synchronized from the perspective of CLK B since CLK B is derived from CLK A
No, CLKB is not synchronized with CLKA, because it's phase depends on a delay and jitter in the clock divider module, which is random.
You're needs to transfer signal from CLKA clock domain to CLKB clock domain.
But as
nctnico mentioned above, there is no guarantee that CLKB is reliable and has proper phase delay, because you're generated it with custom logic instead of using PLL module.
I'm using such approach to get slow clock due to PCB layout issue, because there is no way to use PLL (the signal is connected to a usual fpga pin, so it cannot be feeded to PLL in Altera Cyclone 4), but such clock is not stable. I need a slow clock for a large and slow logic. It can work properly after one compilation, but can stops to work properly after some change in the design (even not related to this part of circuit). I didn't find the way to generate reliable clock with custom logic instead of PLL on Cyclone 4.
In order to transfer signal into CLKB clock domain, you're needs to add a second synchronizer clocked from CLKB.
Something like that: