All right, I've revised this, I'll explain some of these changes at the end of the post:
Bard: College of Psalms
Bards of the College of Psalms devote themselves to the religious practices. They augment their power with radiant energy and that of the divine. Those who follow this path study the practices of Clerics and Paladins in an effort to understand and emulate their power. Whether for good or evil, bards of this college use powers beyond that of the arcane, that of the divine.
Bonus Proficiency
At 3rd level, you gain proficiency with the Religion and Insight skills.
Divine Inspiration
When you choose this college at 3rd level, you have learned how to use your innate well of power to aid your healing. Whenever you cast a healing spell, you may use one use of your Bardic Inspiration. Roll the die, add the number rolled to the hit points healed.
Channel Divinity
By 6th level, whatever source from which you have been granted divine magic has allowed you power similar to that of other holy warriors. You gain one use of the Channel Divinity feature, which recharges at dawn. Choose one cleric domain or paladin oath, and choose two of their Channel Divinity options. You may choose Turn Undead no matter what domain or oath you choose.
Should you choose Turn Undead and/or Arcane Abjuration, they count as Destroy Undead and Arcane Banishment respectively, and scale the same as with the Cleric class.
Holy Seraphin
By 14th level, you have been granted divine power, allowing you to temporarily assume the form of a lesser angel. You may use an action to activate this ability, which lasts for one minute, until you are incapacitated, or you use a bonus action to end it. While in this form you gain the following benefits.
You gain feathered, angelic wings, which grant you a flying speed of 40ft.
You shed bright light in a 10ft radius centered on you, and dim light 10ft beyond that.
Whenever you use a healing ability or cast a healing spell, use maximum number you can roll.
As a bonus action, you may touch a dying creature and stabilize them automatically.
Once you use this ability, you cannot use it again until you finish a long rest.
Explanations:
Divine Inspiration-I saw people suggesting using Song of Rest or just a straight healing Bardic Inspiration. The reason I didn't do those is that I was modeling this ability somewhat of Valor's Combat Inspiration, which uses Inspiration to modify AC and damage, it didn't make sense in my mind to have it be that this doesn't modify something. Even Cutting Words modifies an opposing attack roll.
Channel Divinity-I saw the comments that it was a little to broad, so I scaled it down a little, taking the suggestions to have the bard choose a Domain/Oath. I still wanted to leave Turn Undead accessible, so I didn't restrict it to domains. I do realize that some domains only have one Channel Divinity option, which would force them to take Turn Undead, but I wasn't exactly sure what I could do to balance that.
Holy Seraphin-I can't believe that I studied Bards to make this subclass balanced and missed that they get Magical Secrets at 14th level already! After a bit of thought, I took some inspiration from the paladin capstones (this is less powerful, at least I hope). I think that besides keeping away from MS, this makes a more interesting capstone than Domain/Oath/Circle spells. I think this establishes Psalmist as a support class pretty firmly.