Nothing is wrong, but I wonder how effective it would be, as a batlemage I assume you are good with blades and destruction so training up another combat skill might be pretty redundant also because destruction work nice for ranged attacks.

One interesting use of bows is that you can use them to poison target at a range. This require no skill as you are most interested in the poison effect.
For this use carry a light bow like silver, if you cast a weakness to poison effect you can even use it against undead, zombies has load of health, liches have reflect magic using it dangerous to use magic on them.