If already depleted units qualify as reserves, then yes. To me, they decided to expend their previously beaten units, while everything else they had was dedicated to entrenching elsewhere.
It's not the force that committed in Bakhmut is the one that have issue for Russia, it's the force that they spend earlier on on both North (Kreminna) and South (Vuhledar) that Russia lost cannot be replaced.
Conventional estimate Russia had deployed 1//10 (around 20000 troop) of their force in Bakhmut, pretty sure Russian wouldn't cry about that even if they lost them all.
The issue here is the North and South arm of the earlier Russian Offensive, you know that is the main event when Russia failed to take both objective and didn't try again, and then again like they did with Bakhmut, which mean those force were important to them. So the question really was how many of those force + amour Russian lost before they withdrew.
Western intelligence put that number anywhere from 40 to 60 BTG.
Russia has around 8 to 10 regiments worth of troops around Bakhmut, certainly not a major part of the force. Out of these, 4 on the northern flank are para units, which showed themselves to be a moderately capable force fighting in the south.
The rest are Wagner, and other irregulars in the city itself.
Bakhmut was all but a symbolic target for both side. And weirdly both side don't care about the attrition on this. But what concern Russia is that Ukraine is hiding their best troop , As I reported back in April, 5 of their veteran brigade were not involved in Bakhmut. And they have withdrew another 4 during late April/May for R&R which mean they would be most likely fresh when they get influx from other fighter from other unit for the upcoming offensive coming in late May.
Problem is, would Russia have enough force to stop this? That remain to be seen, but I would say both side make a wrong move in Bakhmut, Ukraine could have saved up 10-15k manpower by not engaging, and Russia could have spare around 20k men for defence if they didn't grind on in Bakhmut, the only question now anyone should ask is, who made a bigger mistake?