@blitzor:

Considering it is only a little increase over the current package, you have to ask yourself, whether do you have urgency to leave your current job? If you can just stay in your current job and have no pressure to leave, then I personally think there is no harm trying to give the negotiation one last try.

The reason is, internal pay raise is going to be hard/minimal, if any. So, this is your only chance to have a better shot.

Instead of dealing with the HR, I would suggest you to connect with your hiring manager instead (which I believe, you have talked with during interview). State your interest and excitement to join, but also ask his guidance on your expectation of the package.

If they don't budge, it means they don't see you are worth your ask, then it is up to you to take it or leave it.
If you think you are worth the ask, then just stay in your current job and find other better opportunity that really value your skills/talent. (Unless you are under pressure to leave your current job)

Just my opinion, hope it helps.

***A