Does "you're" also qualify as a valid contraction for "you were"?

No, you're, i.e. you are (present tense) is different from you were (past tense).

There isn't a common shortening, but it only saves a letter or two!

Contractions are generally flexible enough to transfer to other bases without much confusion:

They're / We're / You're

They've / We've / You've

They'd / We'd / You'd

You can also stack them if you are feeling edgy (and with mixed success):

You shouldn't've

But other than "'d" there isn't a case for adding extra words that fit the truncated part. Just because "were" matches the syntax for "'re" doesn't mean you can drop it into the contraction:

They're going to live here but now they're not

We're going to the show but they sold out

You're my best friend but now you're my enemy

These sentences just don't parse well due to the same word being used for two different meanings without anything but context to distinguish them. The biggest problem is that the context for each is more or less the same:

You were going to live here / You are going to live here

We were going to the show / We are going to the show

You were my best friend / You were my best friend

The could/would problem is similar:

We'd have eaten pizza

But the advantage here is that either one gets close to the intended meaning. Switching between "were" and "are" is too drastic of a change and too hard to see without clarification.