You are letting all your
'book knowledge' confuse what you are actually seeing on the tv show.
Tyrion asked him (in a sacrcastic tone): "So THAT's why you did it? To SAVE lives?"
Tywin telling Tyrion to....
'Go write a song about the Starks'.... was not a REBUKE.
Look up the word
Rebuke in the dictionary. By definition... A Rebuke is a
'harsh criticism'.
It's an expression of sharp disapproval, or criticism.
If his WHOLE INTENTION for talking to Tyrion was to REBUKE him for 'neglecting his duties'... they both would have been TALKING ABOUT SANSA THE WHOLE TIME.

Not discussing the Red Wedding.
If Tyrion was slacking so badly.... why would Tywin mention Sansa for the very first... as he is
walking away from the table? Hmm?
At the 5:20 mark... Tywin says: "Roose Bolton is now Warden of the North... until your son from Sansa comes of age... I believe you still have some work to do on that score"... then immediately gets up & turns his back. (Translation = this convo is done. Get out

but Tyrion keeps the conversation going by discussing Sansa some more)
On the TV Show, it's clear that Tywin (sometimes) likes to listen to Tyrion's
'perspective' on things.
But Tywin mostly talks to Tyrion.... TO FURTHER HIS OWN AGENDA.

He INTENTIONALLY USES Tyrion... TO CARRY OUT HIS 'MASTER PLANS'.
Here's another example...
He tends to 'listen' to Tyrion... in case he has something 'worthwhile' to say... Then BARKS orders at him... to further his own agenda.
On the contrary, I see the plot & characters VERY clearly.
Between the 2 of us.... YOU are the only person who has a very 'hard time'....
SEPARATING... fiction from reality.
Movies like 'Gladiator' & 'Saving Private Ryan' are ALSO based on 'Historical Events'... but I don't need to know all the little details of the 'specific historical events'.... in order to follow the plot, or understand the motivations of the characters.
Because it's..... just a movie.
Not real-life.
There's NO NEED to apply 'real context'.... to a
'WORK of FICTION'.
They are 2 separate things ENTIRELY.
Fiction requires a
'suspension of disbelief'. 
'Real Context' does not... Because it is based on FACTS.... not someone's imagination.
None of this shit is REAL, bruh.

It's just a TV Show.
That's why my understanding of words & concepts like 'murderous', 'sadistic' and 'pragmatic' are CRYSTAL CLEAR.

Because I don't confuse
'how they are applied' or
'what they actually mean' in the real-word..... and in fantasy-land... like you do.
See above. I covered this already.
You are in DENIAL..... He did.
And Oberyn knew it... just like everyone else who doesn't confuse Fiction with Reality knows it.

Except you.
So here's the difference between us....
According to You...
- Tywin 'cares enough' to tell Tyrion to rape Sansa to produce an heir... if it is necessary.
- But he doesn't 'care enough' to command the Mountain to rape Elia Martell.... He just wants her dead.
According to Me....
- Tywin
ORDERS people to RAPE, on his behalf... and he's done it more than ONCE.

(i.e. Tyrion & the Mountain )
- Because Tywin is a sadistic & murderous savage... deep down inside... beneath all his 'dignified' mannerisms.
You posted that Tywin didn't care 'HOW' she died.... while completely ignoring the fact that 'getting raped'... was a 'BIG part of her death'.
But I said: "Tywin DID order the Mountain to rape her.... and kill her"
Hey, Guess what? Lady Olenna just died in the last episode.... and Jamie giving her a bottle of poison... was a 'BIG part of her death'.

Just like the Mountain 'raping' Elia Martell right before she died... was a 'BIG part of her death' also.
"All the pieces matter".... when you are talking about 'Tywin's
motivations.... and whether or not he would have had the
audacity to command a Bannerman to rape a woman, before killing her.