I think there are a lot of really good answers here that make fantastic arguments to Tywin's line of thinking in this situation. Honestly, I'm not sure if we can ever really know what he truly wanted or rather what was more of a priority for Tywin's interests, considering how he handled the Shae situation, letting her testify against Tyrion and because, there may not have been a lot he could of done anyways outside of removing Shae from the situation (and for me, that does seem like icing on the cake, showing Tywin's true feelings)...
However, I also think there is evidence to suggest that Tywin wasn't particularly interested in saving Tyrion's life even before this situation arose.
In fact during season one after Tyrion escapes from the Eyrie & picking up Bronn as his Champion; rallies the hill tribes to his will by making lavish promises to them, then bringing them before Tywin to fulfill these promises (which ALL were to keep Tyrion alive, by the way). Their conversation starts with Tywin bringing up a rumor of Tyrion's demise...
1.08 The Pointy End - Transcript:
TYRION: Uncle. Father.
[TYRION stands before his father. BRONN, SHAGGA, and two other
tribesmen, TIMETT and CHELLA, join him.]
TYWIN: The rumors of your demise were unfounded.
[TYRION looks irked.]
TYRION: Sorry to disappoint you....
Then in the next episode, Tywin seemingly punishes Tyrion for everything Tyrion has put Tywin through and commands Tyrion to lead his hill tribe (and presumably Bronn) in the front of the vanguard!
1.09 Baelor - Transcript:
TYWIN: I do hope your savages are going to be of some use; otherwise
we've wasted good steel on them.
KEVAN: The great hairy one insisted he must have two battleaxes. Heavy
black steel, double-sided.
TYRION: Shagga likes axes.
ANTHOR DAYNE: That's putting it mildly......
[TYRION turns to face ANTHOR, looking surprised to see him.]
TYRION: Ah, the great Anthor Dayne, the twin brother to the Sword of
the Morning. You're a long way from home, Ser; what brings you to the
[ANTHOR raises an eyebrow at TYRION.]
TYWIN: Enough. When the battle commences, you and your wildlings will
be in the vanguard.
[TYRION looks surprised and a little apprehensive.]
TYRION: The vanguard? Me and the tribesmen, on the front lines?
[TYWIN nods softly, and has a slight grin on his face. TYRION still
KEVAN: They do seem rather ferocious.....
ANTHOR DAYNE: You have any experience in combat, Lord Tyrion?
TYRION: [annoyed, to ANTHOR] Perhaps you'd be kind enough to join me.
[to KEVAN] Ferocious is putting it mildly. A Moon Brother stabbed a
Stone Crow over a sausage. Three Stone Crows seized the Moon Brother
and opened his throat. Bronn manage to keep Shagga from chopping off
the dead man's cock, which was fortunate, but even still, Ulf is
demanding blood money, which Shagga and Gunthor refuse to pay.
TYWIN: [sternly, but amused at the same time] When soldiers lack
discipline, the fault lies with their commander.
TYRION: [frustrated] Surely there are ways to have me killed that
would be less detrimental to the war effort!
TYWIN: There'll be no more discussion on the matter.
Now obviously Tyrion never makes it to front lines, but even seeing how easily he was knocked out and trampled upon, along with Tyrion often acknowledging he is not a good fighter, which is showcased in previous events, making deals and getting people to fight for him, it does imply that Tywin knew there was good chance Tyrion wouldn't make it...
1.10 Fire and Blood - Transcript:
TYWIN LANNISTER: They have my son!
In another scene in the next episode, Fire and Blood, viewers see Tywin get rather distressed when Tywin learns that his other [more treasured] son Jaime was captured and learning how Robb Stark laid a trap. I only included this bit to show Tywin's bias, because he never reacted that way toward's Tyrion.
He's also sends Tryion to King's landing to fill in for Hand of King in his absence, which he knows will irk Tyrion and Cersei alike, but not without threatening or warning him not to take his prostitute [Shae] with him. Almost implying that this was his last chance...