The Bishops (and, mostly, priests) were asked (by the State) to determine if someone was a heretic (the charges came from the State, not the Church). That is not an unreasonable thing to ask a Bishop (or, mostly, priests) to do. The State is poorly equipped to make this determination.
Should the Church participate in this process? If they found someone a heretic, the State would likely execute him. The Church could simply refuse to make such a determination, and remove Herself from the whole process.
Would it have mattered? Probably not. In those days, a charge was just about as good as a conviction. By inserting Herself between the charge and the conviction, the Church actually offered HOPE to those charged with heresy.
What a lot of people don’t consider is that not EVERYONE was condemned. Many were exonerated of the charges against them. The Church SAVED many people in the Inquisition. Had the Church categorically refused to participate, those innocent and faithful Catholics would have likely been killed as well.
And the Church offered even heretics the opportunity to repent and be spared death. St. Joan of Arc is one famous example, but the details are not really relevant to this thread. Suffice to say she “recanted” and was spared, but “reverted” and was executed (by the State).
Could the Church lie and say that heretics were not heretics? No, that would be morally wrong (even if it results in a better outcome). For the Church to participate at all, the Bishops (and, mostly, priests) would have to be truthful and call it as they see it.
However, the current stance of the Church is that though they led killings… they did not teach it in the Church?
First, the Church didn’t lead the killings. People were killed by soldiers, not by priests. The priests would prayerfully accompany the condemned and offer them a Crucifix to kiss, as a last-minute act of repentance.
And, yes, the Church has never taught that it is permissible to kill someone because he is a heretic.