The damage bonus is already factored in. The first bow is normally only 38 DMG but got turned into a 48 DMG bow by the +10 bonus.
A 5-shot bow only deals maximum damage if all arrows actually hit their target, which is difficult to achieve if you're not directly in front of the enemy anyway (like shooting a Lynel in the back of the head after the rodeo) ((EDIT: I completely forgot about fused eyeballs, so please disregard that sentence.))
Personally, I use single-shot bows for sniping faraway targets (headshots deal extra damage) and multishot bows to carpet-bomb camps with explosive ammo, to get the most out of the bonuses.
PS: the second one is still better. If at least 2 arrows hit, it already deals 32*2 = 64 DMG whereas the Royal Bow is still only a 38 DMG bow. Plus, elemental affects apply per arrow, so if you for example fuse a shockfruit to an arrow and use a 5-shot bow during rain, you get five electric AoEs stunning and damaging foes nearby.
If you fuse eyeballs to the arrows you’ll have a decent chance of inflicting all that damage though, right? I positively wrecked a flame gleeok yesterday using a 3x lynel bow with ice keese eyeballs.
Oh I completely forgot about those, lol. But you're correct: Especially the elemental Keese Eyeballs are super convenient in this case, as Fire / Ice Gleeoks take extra damage from the opposing element.