So, I'm sure this has been answered somewhere but I want some actual clarification. The bow is not a weapon, therefore, it is used "at the start of battle" to roll one die and add it to the attack score. Then battle happens and with regular battle, another die is rolled for the attack score. Meaning, a character using the bow can roll two dice? Am I correct?

Correct.

This would probably happen during step 2 of battle (p10 base game rules).

If you're going to be picky (and you probably should) the dice are rolled one after another and each can be rerolled using fate (unlike if they were rolled together).