Oath of Music

Instead of spending time selecting every instrument proficiency that exists, I've set it up so you have 10 instrument proficiency choices in the character builder. It's intended that you have proficiency with all instruments but this should cover most of your needs. I don't know how many official instruments there are. Sorry for any inconvenience but this way saves me time.

One handed instruments: tambourines, bells, and other smaller instruments allow you to focus on defense since you can still use a shield. Their smaller designs result in less damage. On the other hand, two-handed instruments [likely] prevent the use of a shield due to the two-hand requirement and therefore offer more offense.

This feature is an important one since it discourages multiclassing with Warlock / Hexblade early on. Nothing wrong with multiclassing! But this provides an opportunity to play a straight-classed paladin and feel like you're not missing out. You won't fall behind for roleplaying and sticking to your character concept.

This feature does not stack with other bardic inspiration, including any from a bard. This is a good ability but a bard's inspiration and this form of inspiration do two different things. This form of inspiration is selfish; you cannot give it to others. So a bard's inspiration will definitely still have a place in the party. Plus, this only ever remains as a d6--while an actual bard's inspiration at 16th level is a d12, double your inspiration. A bard will not be overshadowed.

Adding half of the performance check's result as damage is needed. This paladin is strong because it's not Multi Ability score Dependent--but there's a few things keeping it back. The main issue is the improvised instrument attacks cannot be used with the go-to feats like Great Weapon Master or Polearm Master. In fact, the average damage, even when the instrument is two-handed, is on the lower side, less than a Warlock who just spams Eldritch Blast. For the record, I assume the extra damage from this feature is about +10 because you likely have a +5 CHA which boosts performance and by this point you likely have proficiency in Performance through one way or another which means you have a very good chance of hitting a 20 on the performance check.

+10 damage is equal to one Great Weapon Master attack except you don't take a negative to your attack roll and in fact get a inspiration die. +10 damage once per turn is enough to help out this paladin. And the reason why this is a flat number and not an additional die is because this will not be affected by critical hits. This doesn't become +20 when you crit; only dice are doubled RAW. If you had extra dice that were being doubled on crits you'd then have a barbarian's brutal critical feature, which you do not have. An average of +10 on top of a free d6 inspiration is a good feature, even when limited to a number of uses equal to half your level--all the more reason to straight-class this paladin and avoid multiclassing!

All the Paladin lv 20 features are good. I believe this one is in line with the others.

The restriction on multiple castings of a single spell is included in order to avoid players abusing mechanics in a way I haven't thought of. If there is a situation where a specific spell is super effective, the player cannot continue to use it--so it does limit super-effective spells and incentivizes creative problem solving every round instead of using a crutch of some kind.

The restriction on time is also there to make this feature count whenever a player activates it because it's use it or lose it when it comes to those 5 additional additional spell slots. And while a player is certainly free to use this feature on a non-threatening enemy, they should feel incentivized to only use this when it'll really matter. I assume the DM will account for this great surge in power when planning future combats, so they could be in trouble if they're caught without this when they do need it. All in all, I don't expect this is something that will be used in every combat, even if there's somehow only one combat in an adventuring day. I trust players to know this is a Hail Mary when they absolutely need it.

Is it strong? Yes. It does require 20 levels of straight paladin though. Power is a great incentive to discourage mechanic-driven multiclassing and it rewards good RP. Maybe players won't need to make a pact with some Warlock patron out of the blue. Plus, paladins are still limited by their spell slots. After those five spell slots from 1st to 5th level are used up, there's another 5 rounds where the paladin is going to have to use their regular slots.