Hello @PetrS ,
I actually found that app note shortly after posting this question. You mentioned how my thinking of how the reload points and syncing work was correct, but after reading the app note I think what I am seeing is supposed to happen.
According to Table 3 in the Hardware Synchronization row in AN5303, it seems like the registers in the FTM are updated once the HW trigger is enabled and the rising edge of the trigger is received:

It also says the same thing at the beginning of section 4.3.4 of AN5303:

The configuration that I am working with has the HWSOC, HWINVC, HWOM, and HWWRBUF all set to 1 in FTMx_SYNCONF. According to the reference manual, the observed behavior that I was seeing was indeed correct.

I was confused by these bits earlier which is why I didn't mention them earlier when discussing my configuration.
Just to be clear though, these bits that are set to 1 in the FTMx_SYNCONF are what bypass the FTM loading point so that the new register values take affect immediately?