Many remote controls have a two-phase shutter button just like the camera's own shutter release button. The first detent activates the autofocus and exposure calculation. Pushing it fully home activates the shutter release.
The shutter lag for the second phase should be fairly constant. So if you use this pre-release, and trigger the second phase at the wanted moment, the lag should be constant.
Note that as these cameras have a focal plane shutter, at higher shutter speeds (faster than the flash sync speed) one side of the sensor gets exposed before the other side due to the travel time across the sensor. The time difference may be several milli-seconds, which, however, is not a problem in most normal photographic situations.
Many cameras, including my Canons, have configurable settings for this so-called AF/AE lock function, where you may select which functions are done at pre-release and which at the second final phase. Read the manual and be sure your camera is set up the way you want.