It's been a long time since we (last) saw each other. It's been a long time since I (last) saw you that happy. These are the correct ones. The other two you will hear a lot, though they are not correct.

All of these are fine. It's been a long time since we've seen each other. Said at any point after not seeing each other for a time. It's been a long time since I've seen you that happy. Said after someone is happy and with a long period of not sing the person happy - yet seeing the person often not so happy. It's been a long time since I saw you that happy. Said after a person has been happy - there was a time before that where you did not see the person happy - and now the person is once again not so happy, yet you are talking about the happiness you saw recently, the happiness you saw long before, and the period between them.

It's been a long time since we've seen each other. This is correct. It's been a long time since I've seen you that happy. This is correct. or It's been a long time since we saw each other. Not so good. It's been a long time since I saw you that happy. Not so good. Happiness does not happen in one minute.....it is a state of being, and lasts for a long time, so "since I've seen you that happy" is better.

It's been a long time since we've seen each other. It's been a long time since I've seen you that happy.