1. I'd recommend to first verify the calculation of variance and std deviation by the application and not just rely on those values. It is possible there is a problem in those calculations, that is much more likely than there being loaded dice.

2. I'd recommend to look at dice instead of low luck so the laws of large numbers kicks in faster. Low luck and dice are really no different other than '1' vs 'n' dice being rolled per battle. It could be the way the low luck dice rolls are reported could have an issue as well. If we can verify that 'dice' look to conform to an expected uniform random distribution, then it's a reporting problem of dice stats rather than a problem in the dice randomness.

A single 6 sided toss of a fair die follows a uniform discrete distribution.

Mean of a uniform discrete distribution from the integers a to b is

x¯=(a+b)2=1+63=3.5

The variance is

s2=(b−a+1)2−112=(6−1+1)2−112=3512=2.91677...

The standard deviation is the square root of the variance

s=s2−−√=(b−a+1)2−112−−−−−−−−√=(6−1+1)2−112−−−−−−−−√=3512−−√=1.707825

Wikipedia has a a lot more on this subject Discrete uniform distribution - Wikipedia