Sulfur( not an organic carbon source) in a hypoxic setting like a reactor , on the other hand, does drop alkalinity .
The simplified version of the reaction from the sulfur bacteria's activity uses water ,sulfur and nitrate adding H+ along with SO4 and N2; the H increases acidity.(2 H2O +5 S +6NO3 -----> 3N2 +5SO4 +4H+) There is a small amount of CO2 used and and a small amount of an organic, presumably bacterial mass, produced in the full version.
Also, I think, the sulfur reaction by autotrophic sulfur bacteria converts NO3 to N2 without the equivalent bicarbonate alkalinity put back by NO3( which makes up for the H+ added from ammonia NH3 reduction in the nitrogen cycle).
When NO3 is reduced to N/N2 in the anaerobic phase of the nitrogen cycle it adds bicarbonate alkalinity equivalent to the H previously added by the ammonia whence the NO3 came. So this alk put back step is shorted out by the sulfur reaction and the effect on alkalinity from the ammonia is not offset as it would be in the usual denitrification situation where the bacteria involved are heterotrophic. Just like rmoving nitrate via water changes can lead to alk drops over time .
It's not a reason not to use a sulfur denitrator as dosing extra alk can easily remedy it; just something to monitor and account for in alk dosing.