checking your CoG using AUW and ZFW will ensure that at all stages of flight your aircraft will remain within CoG limits.
Presuming that the non-fuel loading elements remain unchanged,
(a) if the fuel arm is constant (typical bugsmasher unswept plank wing) and the relevant forward/aft limits is/are constant (or only contract with increasing weight), this should be true
(b) if the fuel arm is variable (applies to most larger aircraft) and/or the forward/aft limits expand with weight **, you cannot rely on a simple TOW/ZFW check of CG to satisfy the requirement to stay within the envelope as fuel burns off. You would need to plot a number of points, according to the shape of the fuel line and the envelope limit lines, to satisfy the requirement.
It is not at all uncommon for an aircraft, starting with a loaded CG near to a limit, to have the fuel line move outside, and then back into, the envelope as you burn off fuel.
For a well designed loading system the designer will (should) have taken care of this problem for you such that the pilot stuff remains straightforward .. you can either just use the end point calculations (with the problems being "hidden" within the system) or there will be one or more specific limitations for you to observe to manage the loading problem.
A case of another trap for young players, I guess ... and not generally well understood by the piloting fraternity .. which is why I bore PPRuNe folk with weight and balance stuff from time to time.
So far as the Regulator is concerned, I can recall a Vic/Tas airworthiness requirement put out by Gary S years ago wherein he dictated a check at gross and ZFW as being the mandate for Weight Control Officers in designing and approving Loading Systems.
I argued against that but, in the end, lost the toss. Some discussions with Gary I won, some I lost .. the way such things go. A good lad, though, and still beavering away in aviation in retirement.
** by this I mean that the forward limit moves more forward with increasing weight and/or the aft limit moves more aft with increasing weight.