Using "that" before a preposition

You're right; these are all correct:

Here's a nice recording, which I think you will like listening to.
Here's a nice recording that I think you will like listening to.
Here's a nice recording, to which I think you will like listening.

I cannot adequately explain why your last example is wrong. It may just be that that has to immediately follow the object it refers to. A natural replacement would be one you have already given:

Here's a nice recording that I think you will like listening to.

(This is my personal preference.)

Or even:

Here's a nice recording I think you will like listening to.

However, if you're keen to avoid having the preposition at the end,

Here's a nice recording to which I think you will like listening.

would also be fine. Although which is not usually used for restrictive relative clauses, it is perfectly acceptable: the most important thing is to leave out the comma.

In March, in a discussion about another question, Aaron said to me that "'that' is never a relative pronoun". Aaron gave me a source for this, but I was not convinced.

However, the present example gives me a convincing reason to believe the argument.

In "a nice recording which I think you will like listening to", "which" is a relative pronoun, and the "to" can by moved before it to give "a nice recording to which I think you will like listening": a bit formal, but perfectly grammatical.

However, as Aaron claimed and I now accept, "that" in "a nice recording that I think you will like listening to" is different: it is not a pronoun, and cannot be preceded by a preposition.

According to the theory, it is what linguists call a complementiser (or complementizer) identical to its use in "I think that you will like listening to this".

So at some underlying level, the construction is "which that I think you will like listening to", but a subsequent rule says that at least one of "which" and "that" must be deleted.

Since "that" and "which" actually have different grammar (even though they are apparently interchangeable in the first example), it is no surprise to find that the same transformation cannot be performed on both of them.