I'm a little awkward admitting this. But I tried Norman period re-enactment at university. I once accidentally slammed a very blunt spear butt into a man behind me. He ended up needing three stitches since it split the skin along the eyebrow ridge quite nastily.. It's a myth that blunt bludgeoning weapons don't shed blood. And if you are in battle, you would be stupid to not be equipped as lethally as possible because otherwise you faced the simple fact that someone with a real weapon, in plate armor, isn't going to be easily bludgeoned.

So clearly citing myself isn't legitimate. But as Charles Henry Pearson points out in The Early and Middle Ages of England canon law forbade priests spilling blood.

Richard A. Smith's "Henry Despenser: The Fighting Bishop" points out knights could become bishops and many fought in battle throughout the middle ages and generally did not get prosecuted under canon law.

It's possible some clergy who fought in battle tried to use bludgeons to try and not shed blood, but historically I've not seen record of such largely because the histories don't tend to discuss the "lower echelons" of the armies of the time in much detail. In general the examples you see cited were noblemen and simple fought as bishops as they did previously, in full armour and with whatever weapon they chose, Despenser, Odo and Zouche being prime examples. Zouche was mentioned using a mace but a mace, even blunt, would cave in a skull which would equal bloodshed.

Bishop Hatfield of Durham fought for the English at Crecy with a spiked ball on a chain... Plenty of blood spilled there.

It appears to be tangentially related to canon laws relating to clergy, blood-shedding, and violence. There is a brief mention in Tyerman's God's War: A new history of the Crusades, p75-76, to Bishop Adhemar of Le Puy wielding a mace for this reason, possibly with other clergy.

The fullest treatment of the issue is probably found in Lawrence Duggan's book Armsbearing and the Clergy in the History and Canon Law of Western Christianity, which sadly I don't have to hand, and can only browse snippets of through google books. However it generally suggests that there was a prohibition against violence by clergy, excepting perhaps self-defence, up until the 12th century.

There is a specific prohibition against clerics being involved in judicial affairs resulting in death, and thus mention of 'shedding of blood', in Canon 18 of the Fourth Lateran Council, but it is not as clear as one would think. One should also be aware that 'shedding of blood' may, in some texts, function metonymically for 'violence occasioning death', rather than the very literal sense that D&D has taken it for.