[codingdave]

That does not match my experience. In small companies, I've asked for more and simply gotten it. At large companies, I've asked for more and in most cases they are willing to move the number as long as I don't push the role up to a higher level/band in their HR structure. And my most recent role, they specifically told me that they would need to give me a higher role/title for the money I wanted... I replied, "Great, re-write the offer for a different role, then." They called me back a couple days later with exactly that.

So I'd call their bluff. When they ask if you want the lower offer, say no. I can tell you that most places have recruiters being told to get you in the door for as little as possible... but still to get you in the door.