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You should have let me given you a ride home.
You should have let me give you a ride home.
Your should have let me gave you a ride home.

You should have let me driven you home.
You should have let me drive you home.

You should have let me drove you home.

I'm kind of embarrassed to admit that I'm not sure which of these are actually correct and more importantly which are definitely incorrect. Part of me feels like since I'm speaking of past tense the whole sentence should be past tense but since I've already referenced past tense maybe it's not that important? My inclination is the first sentence of each is correct but I'm not positive, or certain if the others are technically wrong. It's weird when speaking about something in past tense that didn't actually occur.

To form a sentence in the past tense, there is no requirement that every verb be in the past tense. You've already demonstrated this. It's "You should have" not "You should had."

In both of these cases, neither the mood of the sentence, nor the tense matter. The issue with given/give/gave is different. Usually, when one verb follows another verb (and the first verb isn't part of a compound tense, e.g. "I have seen"), the second verb will be in the infinitive. For example, "I want [verb 1] to see [verb 2]."

But your examples are somewhat special. The verb let is always followed by a bare infinitive (the infinitive - to give, to drive - minus the to).

You should have let me give

You should let me give

You let me give

If you had let me give

If you will let me give

Here's a BBC article on verbs using the bare infinitive:

  let + object + infinitive

Like make, see and hear, let is followed by object + bare infinitive. It cannot be followed by verb-ing:

  Let me carry that box of papers for you. It's very heavy.

  Why don't you let him walk home by himself from school now? He's eleven years old after all

Let is also frequently used in the expression let's (let us) to introduce a suggestion. Note that negative sentences with let's can be formed in two possible ways:

  Let's finish the video tomorrow, shall we? I'm tired and I want to go to bed.

  Let's not be late home tonight. It's Monday tomorrow after all.

  Don't let's get too stressed about this. I know the car is damaged, but it's only a piece of metal.

We do not normally use let in the passive voice.
http://www.bbc.co.uk/worldservice/learningenglish/youmeus/learnit/learnitv214.shtml