Bolts of Soothing

While the absence of this spell from the Wizard list helps prevent some abuse, It's still going to combine with sneak attack in a rather extreme way. Even without sneak attacking, sharpshooter's +10 bonus is a pretty significant chunk of healing. The action economy cost of "spending" your attack on healing does somewhat ameliorate the extreme outcomes this could produce, it's still susceptible to the same kind of problems that Healing Spirit is.

Hm, I didn't consider sneak attack or sharpshooter when adding the damage bonus to healing. That could be abused with a Rogue multiclass, although that would require advantage for each shot. However, my thinking with this spell is that because it requires the user to hit the creature receiving healing, it's healing potential is at least a little checked. Additionally, unlike Healing Spirit, this would be limited to a max of two targets per turn barring a hefty Fighter multiclass.

Sneak attack doesn't require advantage, just having an ally adjacent to the target is sufficient (and some rogue archetypes don't even require that). The real "build" here is probably a Sharpshooter/Crossbow expert on a Ranger with the Archery Fighting Style. Extra attack with Bonus action hand crossbow means three attacks per round; and unlike healing spirit they can be focused on a single character.

Following on that build, I notice upcasting this spell adds a d4 to the healing and the normal damage of attacks. That means you now have to compare it to Flame Arrows and Elemental Weapon. Compared to those spells, the lack of concentration on Bolts of Soothing puts it pretty strongly ahead. A single concentration-free buff that passively increases damage as well as granting a healing trick. This is probably doing too much.

Ah, true about how it doesn't require advantage, but only depending on how you read it. The rule reads: "You don't need advantage on the Attack roll if another enemy of the target is within 5 feet of it, that enemy isn't Incapacitated, and you don't have disadvantage on the Attack roll." This implies that you are one enemy of the target, and your ally is the second. This could be read many different ways depending on the DM. Such as: The target must be 5 feet away from an enemy, or the target must be your enemy in general, the assisting creature must be your ally, etc. Some may say Sneak Attack never applies even, since you're never really taking advantage of an opportune moment/ drop in enemy guard, and your allies ideally know what you're doing and it's not a surprise. 

Also, the Bonus Action hand crossbow is moot, since it is a second projectile weapon used, this spell would only apply to one. To have two weapons under this spell would require two casts of the spell.

And finally, compared to Flame Arrows and Elemental Weapon, it is more powerful in certain situations, but to gain those damage benefits it must be upgraded to a 3rd level spell at least, matching them. Being non-concentration is its trade off for only lasting one minute, unlike those others which can last an hour. Plus, the latter gives a to-hit bonus and you can choose your damage type, and the former is already considered non-competitive with other spells of its level. The damage scaling IS more than Elemental Weapon, but Rangers will only get 5th level slots so it isn't too much worse for the main damage dealers using this spell.

Crossbow Expert doesn't require you to use two weapons, per Sage Advice

Does Crossbow Expert let you fire a hand crossbow and then fire it again as a bonus action? It does! Take a look at the feat’s third benefit. It says you can attack with a hand crossbow as a bonus action when you use the Attack action to attack with a one-handed weapon. A hand crossbow is a one-handed weapon, so it can, indeed, be used for both attacks, assuming you have a hand free to load the hand crossbow between the two attacks.

The big advantage a lack of concentration grants is that it can stack with those other spells. A ranger can combine Hunter's Mark with your spell. Likewise paladins with Divine Favor. While this shouldn't increase the healing effect, the overall damage is still unusually high. This effect is amplified on multi-attackers, which is why the workings of Crossbow Expert are so important.