If its due to liver dump, then it should improve as the fat stores deplete from the LC diet. if the levels still creep up in the morning, then it may be what was eaten before bedtime doing the biz. There is a third possibility, and that is a nightime hypo occurring. Normally i would say that oral meds should not produce one, but a reading of 5.4 could drop lower due to the Gliclazide, and lead to a hypo with the liver taking emergency steps to protect. This is more of a problem for insulin Dependants, but Gliclazide and LCHF could have the potential for it too.
A nightime hypo is difficult to detect when asleep. If he wakes up sweating then that might be a sign, or a headache / muzzy head in the morning. Other than that, and in the absence of a CGM meter (constant monitor) then waking him up in the early hours and testing is the usual way to catch one.
One last question, What meter is he using? Some meters read high due to either test strip variances, or their calibration, so a 5.4 on the meter could actually be physically lower.