Okay, a stream of conciousness as I read through.
Interesting fluff, but I don't really have an idea from it how they play besides beign "fighters with magic". At the table, how do they play different than paladins, eldritch knights, etc. What's their niche mechanically?
d10 HD and half casting - keep in mind if they obsolete the Eldritch Knight
Int based features/casting - good, and also not too easy to abuse with multiclassing.
Arcane Sense - sure. No ritual casting which is how others sometimes do this.
Supernatural Prowess - this is very similar to the first kicker for the Samurai fighter subclass in UA, but when they published it in XGtE, it's now a bonus action to use and only lasts for the current turn. Having this as a free action that lasts for two turns is out of line with that. If this is their only major power boost it's probably okay, but considering that they also get half casting that's likely not the case.
Spellcasting - spells known is about the same as a sorcerer of the same level, which seems like too many options of a half caster. I'd suggest moving to the same spells known progression as the Ranger. (Note, I'm saying this before seeing the spell list - these are thoughts as I read through.)
Empowered Weapon vs. Hex/Hunters mark - better action economy in not needing a bonus action to change targets. Level increases damage instead of duration. Damage starts lower, and only affects one weapon (so doesn't boost offhand weapon or spell attack damage). Gives a good reason to be in melee. Doesn't take up your concentration slot - this is a biggie. Seems fine, though the damage with high level slots may be too low to make them used.
Hmm, looking at Divine Smite plus the new Warlock invocation, total damage bonus for a slot is around d8+d8/level. With extra attack assume one hit per round and four round of attacks. So a 3rd for you is 4d8, right on target. 4th is 4d10 vs. 5d8 - good. 4d12 vs. 6d8. Okay. They get better nova, you get more hits then I suggest (about 65% on average, before Supernatual Prowess). So I take it back, the damage looks good.
Magic Resistance - very reasonable for the single defensive power.
Use Magic Device - this is fairly unique in 5e, and something that seems a bit out thematically - why are you a master of magic items?
Master Prowess - very powerful, but hey, capstones are supposed to be. This is basically advantage on all attacks all day at most tables.
Calling of the Spellblade:
Duelist's Prowess - again, reminds me of the UA Samurai ability grantign disadvantage, and the final XGtE version the defensive aspects are just 5 temp HPs (10 at 10th, 15 at 15th). I'd drop it down to that. Also, granting both advantage AND increasing crit range works too well together. Extra attack, 2 rounds, and a 19% chance to crit per attack. Often with extra weapon dice due to Empowered Weapon which also double. Just drop the crit range boost.
Spell Channelling - Full casting and a weapon attack. Eldritch Knights get a spell and a bonus action attack at 18th, and a cantrip and a weapon attack at 7th. Granting this at 3rd is a great deal more powerful, even if limited to single target spells. This really ups the damage per round, and you're already doing that with Empowered Weapon. Plus you can hit more with Supernatual Prowess. It feels like there are too many things that stack - each might be fine in isolation but taken together are too much.
Improved Spell Channelling - okay, use the weapon attack roll for the spell attack roll - good by me in a vaccuum. But in practice it's more powerful when you consider Supernatural Prowess.
Superior Spell Channelling - a free reroll ... when you can already get advantage very easily. And it's delivering a weapon attack, and a spell, and possibly Empowered Weapon rider damage. It's just too good at delivering hits and increasing the damage for them.
The whole spell channeling section: It's too much with some other aspects of the class. I'd rebalance it. Here's one suggestion on how to do it: Change Spell Channelling to be INT times per long rest and cap the spell level used to your current Empower Weapon slot used. Change Superior Spell Channelling to only reroll a single d20, not "the attack" which if used with advantage is 2d20.
While Calling of the Vigilante is next, I have only a limited time right now so I'm going to jump to spells and then some conclusions.
Spell List - seems good. I was worried about it being really long. It is a greatest hits of damage and self-buff, but you want to use your magic to do those for how you described the thematics.
Conclusion:
Looking at how WotC nerfed the Samurai's ability that was very similar to Supernatual Prowess, I'd strongly suggest the same thing - bonus action to activate and only lasts for the current round. You've got all of the casting as well, the fact that you are just "on par" with a Fighter - the standard for melee combat - seems reasonable. WotC changed it for a reason.
I mentioned several changes for Spellblade, I think they hold up.
All in all, this is still more powerful that then Eldritch Knight - they get features like the main parts of fighter subclasses AND casting, while the EK gives up Superiority Dice or Crit Range (or the Samurai ability) in order to get casting, and that casting is inferior. To balance the greater arcane focus I'd suggest dropping the HP/hit dice down from d10s to d8s.
I look forward to finish reading the other Callings, the class seems flavorable.
