Ursarctic Mitsurugi

Strengths

The deck can very effectively kill a lot of Extra Deck-based strategies going first, including Ryzeal and Maliss (assuming you don't get hand trapped). Usually you can also kill Fusion and Synchro-based decks if you can properly resolve Ursarctic Polar Star (which happens more often than you'd think).

Unlike other Ursarctic builds, if you are stopped mid-combo, you might be able to pivot towards your Mitsurugi engine which does quite well in the grind game, through Futsu no Mitama no Mitsurugi's constant Special Summons from the Graveyard, a board wipe in Ame no Murakumo no Mitsurugi, and even an omni-negate through Mitsurugi Great Purification.

The Mitsurugi engine also provides some more benefits: the deck is slightly less vulnerable to the threat of Droll since Prayers is a Quick-Play Spell that you can activate during the Draw Phase, your Tributes actually do something now, Nibiru isn't as much of a threat as it used to be, and on top of that, it only needs one card to get started: any way to access Habakiri. 

Additionally, due to the Levels of the Mitsurugi monsters, we have access to Crimson Dragon and Ultimaya Tzolkin, additional extenders that you can utilize to access negates in Crystal Wing Synchro Dragon or Hot Red Dragon Archfiend Abyss. 

There's also some synergy in Ursarctic Grand Chariot and the Mitsurugi monsters - although resolving Grand Chariot's negation effect is unlikely. 

Weaknesses

The deck is still highly susceptible to Droll and will almost certainly just stop working. At most you might be able to get to Futsu and just go for a grind game.

The Mulcharmy cards are also absolutely devastating. Purulia and Fuwalos are serious threats to the Deck and there's barely a good way to play around it if you only draw into your Ursarctic cards.

It's also quite hard to get some of your resources back if they're banished. Dimension Shifter will technically not stop Kusanagi or any of the other Mitsurugi monsters' effects, sure, but it still hurts a lot as you can't get back those resources once they're banished.

The two engines also don't really "feed into each other" and often times you can actually just use one engine and leave the other out of the picture until you can actually use it later. Ursarctic technically has a way to maybe get to your Mitsurugi stuff through Ursarctic Radiation, but the chances are usually quite low. 

Also... you're still playing Ursarctic. All of the other threats that the deck suffers from are still present in some way. The deck is still quite susceptible to interruptions like Ash, Imperm, Veiler, Belle, Impulse, Purge, the list goes on. That's why our non-engine cards are meant to help break boards (Super Poly & Fenrir) OR stop those hand traps. You might think that we could just run 12 hand traps/staples, but Ursarctic's inconsistency and neediness for other cards (there's no 1 card combos) makes this kind of weird non-engine, in my opinion, kind of necessary.

Tips

• Use your Ursarctic cards first if you have any of the necessary pieces to start the basic Mitsurugi combo. The Mitsurugi engine is much stronger, and baiting out common hand traps with the Ursarctic engine can allow you to give you some sort of board to work with.
• Activate Mitsurugi Prayers during your Draw Phase in order to play around Droll - just a little, anyway. 
• If you have a Habakiri, Mitsurugi Ritual, and an Ursarctic in hand, you can Tribute the Habakiri to add Murakumo - this can allow you to access Futsu.
• Use Habakiri's effect to Special Summon (ideally Saji or Aramasa) BEFORE you Tribute it with an Ursarctic monster.
• If you have nothing good to add to your hand off of your Mitsurugi cards, add Habakiri. It's a great tribute and can be used as discard fodder for some of your other cards if need be.
• If you have a Level 4 Mitsurugi monster, a Level 7 Ursarctic, and also a Level 8 Ursarctic Tuner, Synchro Summon Crimson Dragon first, add Synchro Rumble, then revive your Ursarctic Tuner. If you control Crimson Dragon, avoid going into your Ursarctic Synchros unless absolutely necessary. Opt for a Synchro 9 so you can Special Summon Hot Red Dragon Archfiend Abyss. 
• Shenshen is TERRIBLE unless you're playing against Mitsurugi or pretty much almost any other deck. Don't risk a bad Maliss matchup and opt for Wolfrayet (if you know what they're playing and it isn't Maliss then just opt for Shenshen)
• Special Summon Fenrir before you do anything else. He's the best bait and utility card we have in the deck.
• Avoid using Droll while your Mitsurugi cards are still live. If you don't want them to access certain cards, you have tools to stop that. 

Tech Options

• Azamina Engine - A 6-card engine I heavily considered and tried out. Consisting of just 2 WANTED: Seeker of Sinful Spoils, 2 Diabellstar the Black Witch, 1 Deception of the Sinful Spoils, and 1 The Hallowed Azamina, it synergizes with the deck a little more than you'd think it would. Diabellstar can be used as material for Ursarctic plays, and Deception can Tribute a Mitsurugi monster to give you access to Hallowed. Another omni-negate doesn't hurt, after all. Downsides are losing some needed hand advantage due to Diabellstar, and also losing some quite valuable non-engine space. 
• Diviner of the Herald - A great way to get to Polari with any other Level 7 monster AND search out Habakiri or Mitsurugi Ritual. While for a time I tried using the card, it is unfortunately really susceptible to Imperm and Veiler. I took it out after some testing with it, since I figured Diviner could have better been used as another Level 7 or higher Tribute that's guaranteed instead of just accessible.
• Sakitama & Aratama - I'll be honest, I've never tried running these, but they could prove useful to help you go into King of the Feral Imps in order to begin your Mitsurugi plays (or just go into Bagooska if you get Drolled...). Downsides are that they can't be Tributed, and also the fact that the engine isn't exactly 
• Vermillion Dragon Mech and Navy Dragon Mech - These are amazing going second cards, but I feel like the existing Extra Deck is fine as is - the Synchro 9s we have are relatively strong and should do the trick. 
• Kuibelt the Blade Dragon - It's a pop that can be used on your opponent's turn with Crimson Dragon - I feel like the negate off of Hot Red Dragon Archfiend Abyss is a little better though.

That's all! If you have any ideas that'll help me improve the deck, let me know.