How much room do dealers have to haggle with?

I’m looking at a new car that ticks all the boxes for my needs and wants. I’d be buying with PCP.

Looking at deals online, if I pay £2000 more in my deposit than I want, I can get the payments to where I’d really like them to be. These deals include their ‘special discounts’ and ‘dealer contributions’.

I’ve bought new cars before and was always pretty happy with the deals I got so have never really done the negotiation game.

My question is, how likely am I going to be able to get them to drop the price by £2k to be where I want, given their ‘deals’ already have ‘discounts and contributions’?