What to add to the batter of the cake to avoid hardening when the gluten formation can't be avoided?
Milk won't help you - it's mostly water, and gluten develops from flour (more accurately, specific proteins in flour) and water.

The way to reduce gluten development is to incorporate more fat into the batter. Lipids are hydrophobic and will prevent further hydration of the glutenin.

Using a lower-protein flour will also help. If you're not already using cake flour, the reason it's called cake flour is because of the lower protein content.

That being said, have you actually tried leaving the batter coarse? Just because the batter is lumpy does not mean that the cake will have big lumps. The entire mixture is wet, so unless you leave huge lumps of dry flour in the batter, you won't end up with a lumpy cake. There's a difference between "don't overmix" and "don't mix" - you're supposed to mix enough to incorporate, just don't try homogenize it.