I've been playing Tyrant Oath for months, so I can give you some tested thoughts. You can also search the forum for a couple of threads I've posted, one with a chalice-based Tyrant Oath, and the other without.
First of all, I think the version without chalice is a bit superior, even if this is dependent on the metagame: if you get double tyrant and a chalice in play you've simply won, but that doesn't happen often, or fast. Having just a Tyrant and a chalice is nice, but doesn't win the game immediately, even if most of the time could slow the opponent enough to win. Usually, I've found the cunning-freeze finisher the best one, with control-beatdown being the Plan-B, in particular against decks where I side in chalice (because, OF COURSE, they are in the sideboard for certain matches like TPS); the problem is that chalice is not the best card against every deck, hence my decision to keep it as a sideboard choice.
From your list, I see you run just TOO MANY tropicals! You have just 4 green spells! One or at worst two tropicals are definitely enough.
Some words about the cards you've told:
Exhaustion: a 3 mana lock. This card commands another turn in a deck that already decides what you get to keep in play. (or a 2 mana time walk with grand arbitor in play)
Crap, really. It is just a win more or a do-nothing. It really doesn't serve any role.
Cunning wish: Looking for a stifle to not get stormed out?
I play with an handful of duress, so I don't fear trickbind too much; with chalice, when you combo out, you can also set chalice at 1 or 2 if you fear stifle or trickbind. Even if you set them at 2, your brainfreeze will be countered, but all its copies will not, and you'll win anyway. Resetting chalices at 2 or 5 when comboing out to protect cunning against drain/force is a good thing to do also.
Personal tutor: I find the deck lacks go-get'em cards, but a 1 mana casting cost is clearly inferior in a deck that PLANS to play a 1Chalice when the card comes up.
Mystical is very good, but I don't think Personal tutor is really worth it with its sorcery speed. After all your win card is an enchantment, not an instant or sorcery; decks which have sorcery/instant winners don't run it: why you should?
Brainstorm: I replaced these with impulse because i kept drawing them with 1Chalice out. a clear choice, only drawback i find with impulse is now the deck lacks any way to put those 2 tidespouts in your hand, back on top.
This is a difficult choice: I've tried to play without brainstorm, but it's not the same. The problem is not putting back Tyrants: the odds you get both in hand is little, and after all you only need one in play to win (I run only 1 maindeck in the chalice-less version after all); the problem is adjusting your hands and protecting your bombs. Overall, I think brainstorm is still worth inclusion, even if you want to play chalice@1, at least as a 3x.
Grand Arbiter Augustin IV: A new card i added recently and replaced with gaeas blessing. but suddenly this makes my chalice my worst enemy, I cant flashback Deep Analysis goodbye impulse, echoing truth, time walk. Was considering just adding a third tidespout in his place.
First: lowering your two mana cc spells does not put them under chalice@1: they still are cc2 spells even if you pay only 1 for them.
However, GAA is just a win more, and not even a good one: if you flip with oath it first instead of Tyrant you put yourself a whole turn behind; ok, if you get this AND Tyrant that's nice, but not an auto-win, and two Tyrants would have been. Just cut it. Ah, and don't run 3 Tyrants, 2 are even too much (but a fair choice in a chalice build). You could think about running 2+ Tyrants only if you plan to play one or more Show and Tell, and I could do that only in a metagame full of chalice@2 and meddlings.
Deep Analysis: wowza! 2 mana, 3 life, 2 cards. Who can complain!? except when its in your hand... and have no way to pitch it to your GY except spending 4 mana on your main phase to draw 2 cards.
Deep Analysis is good with TfK, and it is not so hard to cast after a drain, and it's simply awesome when you flip it with oath, giving you the same card advantage as an ancestral (+2, as an ancestral if you consider the latter needs you to lose the ancestral itself from hand) and granting you to have one or two spells to cast as you flip Tyrant, making you bouncing cards even if you was empty-handed. Even if they kill the first Tyrant, you rebuild your hand and prepare the win with the second Tyrant and a regrowth hand.
Hurkyl's Recall: I think i need to find a slot for this, its a lot of bounce and mana when a tidespout is in play.
Your major fear is Chalice@2. Don't play Hurkyl. If you want a bounce, play something among Rebuild, Repeal, Rushing or Cunning for something. Even chain of vapor is nice.
Yawg's will: some people say this doesent belong in oath, I tell those people to test the deck 100 times and let me know how many games were won by casting it, was it more then 5? Then what game winning card do you plan on replacing with it? I like it, it works well with no gaeas.
YagWin is another big win condition: even if risky, you can win milling your deck, reclaiming Ywin and winning playing your grave for a lethal freeze (if you run at least a cunning). Sometimes instead, it juts gives you SO much advantage. You could think about running black or not, but if you run black, YWin is an autoinclusion. BTW, I see you don't run Krosan Reclamation. That's a deadly card with Ywin, make you shuffle back good cards you've milled, blocks some cards from your opponent's graveyard, shuffles away some of his top-deck tutor, shuffle backs a tyrant discarded with Thirst, and prevent you from losing if you are so unlucky to flip Tyrant as one of you very last cards.
Lat-Nam's Legacy
Slow, color-intensive, card disadvantage. Just no.
