A few things impact this...
You're meant to use this a lot.
Your Mantle of Inspiration is a parallel to a Lore Bard's cutting words, a Valor Bard's Combat Inspiration, and so on. This is your sub-class's 'fancy trick' with your Bardic Inspiration, and every sub-class gets a LOT of use out of their trick.
Having played a Lore Bard, I can freely say that I tended to get more use out of my Cutting Words than I did out of normal expenditures of Bardic Inspiration. The ability to reduce the total damage of AoE spells is fantastic, or make that dragon whiff when it takes a swing at my nearly-dead ally.
Burning your Reaction is not necessarily a 'low-cost,' and becomes less so as you level up.
Burning your Reaction means no opportunity attacks and it deprives several classes of special features they have (like a Battlemaster's Parry, or an Arcane Caster's Shield)
You seem to be getting an inordinately large number of Short Rests.
A Short Rest is a full hour of rest. The standard adventuring day as proscribed by the DMG is six to eight encounters, with two short rests interspersed in there. You should not have all 5 uses of your Bardic Inspiration for every combat.
Excessive Short Rests tends to screw with the balance of 5E (such as making Warlocks ridiculously powerful)
There are many things Mantle of Inspiration doesn't help against.
8 temporary HP isn't worth crap if a character gets hit with Hold Person and gets assailed with a flurry of auto-crits. But BI to boost their Saving Throw may save them from failing that save in the first place. It's better, mathematically, to successfully save vs. Fireball than to have 8 temp hp to help tank the damage. Temp HP is useless for aiding skill checks. And temp HP is only so useful if your party is having a devil of a time hitting their target.
In a typical combat encounter...8 temp HP and the chance to relocate for a Reaction is pretty powerful. But normal use of Bardic Inspiration gives your fellow players an "Oh Crap" fallback. It can let PCs succeed when they might not otherwise (such as letting the poor noisy Fighter succeed on a Stealth check).
In short...Mantle of Inspiration does precisely one thing, and does it well. This is definitely powerful in combat. But traditional Bardic Inspiration is the ultimate multitool. Need to succeed on a skill check? Need to hit with that attack? Need to succeed on a Save? Need to break a grapple before the Grell drops you off a cliff? There ya go.
Finally, Bardic Inspiration is something you can pre-load allies with. Once inspired, they can opt to use it at any time in the next 10 minutes. So if you're about to go into a boss fight, it's not a bad idea to go ahead and pass out Inspiration, leaving you clear to use your Bonus Action for other things and granting your party each one 'probably get out of trouble free' card..
Summary
If you are doing nothing but relatively straightforward combat against enemies that aren't packing Save or Suck abilities or high-damage save-based spells, versus enemies your party can reliably hit...then yes, Mantle of Inspiration will probably get used a lot. You shouldn't be able to use it as often as you do (see the bit on a your surfeit of Short Rests), but it's a good feature.
But MoI is nearly useless outside of combat and is mathematically inferior once you start dealing with most abilities that call for Saving Throws. Especially the ones that burden a character with debuffs.
Mostly balanced except for Dance of the Dead
Overall this subclass doesn't appear to be too unbalanced. Playtesting it as you are probably gives you a better idea than any of us would have purely from reading the rules.
Grim Secrets
Part one of this feature seems fine. Though no other college gain a cantrip at this level I don't feel this is fundamentally different to any of the others.
Part two is harder to guess at as it grants an ongoing improvement rather than a static benefit. To fully know if this is OP I would have to compare all necromancy spells to all bard spells and see if this adds a significant amount of power. I feel as though it may be a little over powered compared to the existing subclasses.
Potentially consider moving this to the sixth level to mirror Additional Magical Secrets of the Lore Bard.
Haunted Eyes
Advantage against fear is common enough from many sources so not a big deal. Necrotic resistance is good but situational so not game breaking.
Dance of the Dead
This feature seems to be both over powered and goes against the normal design principles. For reference the other subclasses gain at this level:
- Additional Magical Secrets (Lore), two spells of your choice
- Extra Attack (Valor/Swords), as it says
- Mantle of Majesty (Glamour), Command as bonus action for 1 minute. Once per long rest
- Fool's Insight (Satire), detect thought CHA mod per long rest
- Mantle of Whispers (Whispers), situational disguise
This feature also uses Bardic Inspiration in a way that no other subclass uses them, more akin to the way superiority die are used than bardic inspiration.
This is the feature that needs the most work and I would consider scrapping entirely for something different.
Rule the still heart
This feature is fine, it could actually be considered weak since it only affects one creature per long rest and they get a saving (possibly with advantage).
Suggestions
This isn't too bad as a first attempt of the subclass. I would suggest re-doing dance of the dead. Potentially replacing it entirely. Allow me to propose some features for you, not sure if it suits 6th or 14th level better though.
Grave Humour
When an Undead Creature under your control, that you can see within 60 feet of you makes an attack roll, an ability check or saving throw, you can expend one of your uses of Bardic Inspiration, rolling a Bardic Inspiration die and adding the number rolled to the Undead's result. You must decide before the DM says whether the roll succeeds or fails.
Puppet Master - credit to Doc in comments
When you use the Attack action on your turn, if a creature you control can see you, it can use its reaction to make a melee attack. Only one creature you control can do this per round.
Best Answer
Sneak Attack is better than you think
In order to land a sneak attack, the rogue needs to be making an attack with advantage, or the target needs to have another enemy within 5 feet (and that enemy isn't incapacitated, and you don't have disadvantage), while you are using either a finesse or ranged weapon.
This sounds kind of tricky, but let's look a little closer. Firstly, a rogue is almost always going to be using a finesse or ranged weapon. Rogues often have a lot of dex so finesse is great for them, and ranged weapons will keep the rogue out of harm.
In a party there is usually at least 1 other melee character, so the rogue will always have a guaranteed source of sneak attacks just by hanging around them.
The rogue can also use their bonus action to hide (rogues often have high dex, so hiding is no big deal), giving advantage and sneak attack. The ease of accessing both advantage and sneak attacks is what powers many rogues.
In combat a rogue will almost always sneak attack for every single attack. What's more, the rogue could even do this more than once in a round, by using opportunity attacks. Even if the rogue didn't have sneak attack, they would probably still be hiding to gain advantage, and working together with other melee characters.
Psychic Blades is worse than you think
While the rogue is out there spamming sneak attacks, the bard is a lot more limited. Firstly, they are limited by their number of bardic inspiration die. With 20 charisma the bard only has 5 uses per rest. That's great if you have 1 fight per rest, but otherwise it's extremely limiting. What's more, if you use your inspiration die for damage, you can't use them for, you know, inspiring people.
Psychic Blades also does not scale as nicely as sneak attack, which isn't a big problem but it will mean that half the time it's dealing less damage.
Bards don't have a lot of feature support for melee combat either. They don't have cunning actions like rogues, and they have to split their attributes between cha and str/dex to be effective. That tax and lack of feature support makes them less effective. Even if they had access to sneak attack, rogue would be superior.
While psychic damage type isn't commonly resisted, a rogue can equip a magic weapon and bypass most physical resistances. By the time this becomes a problem, the rogue probably already has the solution.