What is the tense of "I would have been happy to..."?

There is a tense problem in your original sentence

I would have been happy

refers to something in the past. I would have given you the money, but I did not (because you did not give me a date).

if you would give me a date to leave

refers to something that has not yet happened (you have not given me a date, I'm waiting for it)

The two parts cannot go together as they are, so either you put the dependent clause in the past:

I would have been happy to let you use the money, if you had given me a date to leave

or you put the independent clause in the future

I will be happy to let you use the money, if you would give me a date to leave

The clause "I would have been happy to let you use the money" is an example of a Modal Perfect construction. You can see the modal 'would' and the perfective "have + -en".

The meaning is 'past + counterfactual' -- we use it to discuss something that was possible in the past but did not happen.

To combine it correctly with the second clause see the instruction from Nico. You need to adjust something to get at the meaning you want.

This tense is perfect conditional, because the formation is :Would+have+past participle .For instance, I would have travelled abroad but I hadn't had passport.