What does the British idiom "taking the piss" mean?

That particular usage of "taking the piss", actually means "taking advantage of".

When said from the receiver's perspective it would be akin to "You have got to be joking!", "Are you taking the mick[ey]?", "Are you taking the piss?" - All enquiries as to whether they are being made fun out of, the more usual meaning, but with the implication that is not the case, and they are being taken advantage of.

In this context, said from the side of the entity taking advantage, it says "we will be taking the piss", we will be taking advantage of the other entity.

Particularly used where terms and conditions are particularly harsh, or the client is being screwed for far too much money.

"Taking the piss" simply means "making fun of" in its usual context. It's a very common expression within Britain, but probably rare elsewhere. ("Taking the mickey" is a similar, slightly sanitised version, that perhaps is used elsewhere?)

The Wikipedia page gives a fairly good overview.

Re the second part of your question: yes, there are many many many idiomatic uses of piss in British English! Just off the top of my head:

• piss - urine, to urinate. (The main literal meaning, of course.)
• take the piss out of (someone/something) - to make fun of.
• pissed - drunk, roughly equivalent to hammered in AmE.
• on the piss - out drinking, similar to on the town, on the tiles.
• piss (someone) off - to offend, irritate, anger someone. (Hence: pissed off = angry, closely equivalent to AmE pissed.)
• Piss off! - Go away! (Milder analogue of Fuck off!)
• piss about (or around) - to mess around, do things that aren't really worthwhile.

More suggestions welcome in comments...

My understanding of this phrase breaks down to:

• "Are you taking the piss [out of me]?"
• "I'm just taking the piss [out of you]."

"Piss" in this interpretation could mean something like confidence (similar to the phrase "piss and vinegar"). In other words:

• "Are you trying to make a fool of me?"
• "Are you trying to cut me down a notch?"
• Or in more modern parlance, "are you trolling?"

The dialogue you describe doesn't really make sense for that use of the phrase. "We'll be taking the piss" reads more as "we'll have the last laugh" to me.