First of all if you haven't already done so, remove the disputed image from your website immediately.
You don't mention anything about the photo which is in dispute or how long they are claiming it has been in use, but I think it's fair to say that a four figure sum is likely to be wholly disproportionate to any alleged infringement. That is not say that their claim is without merit, but if the matter did go to court, the most the claimant could expect to receive in normal damages is the amount you would have been charged in licence fees if you had gone down that route in the first place. So, if you know the name the picture agency which is offering this image, go to their site and see if you can find the disputed image, and make a note of the fee which would be appropriate for the type of use which is complained about. Generally images are licensed under two main systems: royalty-free or rights-managed. You should read up on these terms if you need to but, put simply, a royalty free deal means you pay a one-off fee which covers use of the image for a fixed length of time (for instance, a year) for a declared number of page-views. Rights managed means that you get a licence which is more tailored to your specific requirements and may work out cheaper, especially if you only need the image for a short time or for a single publication print run. Usually this will entail a smaller fixed fee plus royalties payable based on actual consumption. Say you wanted an image to go on the packaging of a product, the rights-managed option would usually be the better way to go.
Once you have determined what fee is appropriate for your needs, multiply it by the period the image was in use. This figure should then form the basis of your counter-offer. It is, realistically, what the claimant might get if he successfully took you to court. In other words the judicial process would put him back in the position he would have been in if a normal licence had been obtained. The civil law does not, ordinarily, seek to punish, merely to provide restitution. However there are circumstances in which a claimant can allege that any infringement has been particularly egregious (for example it continued even after the alleged infringer had been notified of the claim, or the infringer had deliberately removed copyright notices from the original image), and if he manages to convince the court about this, it can lead to additional damages being awarded. Depending on the type of court in which the claim was heard, the claimant might or might not be able to ask the court to also award him his legal costs. If the small claims track of the Intellectual Property Enterprise Court (IPEC) is used (it's quicker and simpler) then legal costs cannot usually be awarded against a defendant.
However, assuming this doesn't go to court, the rights owner in this case will still have to pay both PicRights (probably a percentage of anything they recover) and the firm of solicitors which sent the letter, and so he is unlikely to accept your counter-offer unless you are prepared to re-imburse some of these costs. I suggest that you add about 10 - 15% to the licence fee amount and make that your counter-offer to settle the matter. Be aware that the claimant won't want to go to court any more than you do, however much his representatives may bluster.
Unless there are other circumstances which you haven't mentioned, you should stand your ground and not pay the full amount demanded. This sort of activity is known as copyright trolling
and is generally deprecated by the courts. However if you feel there may be other grounds which make it highly likely that additional damages may be in contention, you should probably speak to a solicitor. You don't not need to worry about their deadline which is largely there to panic you into paying quickly to make the issue go away. A court would not entertain any additional costs being added to the claim just because the claimant threatened that a delay would increase the costs, and it would view three or four days to settle the matter as wholly unreasonable behaviour. Also be wary about signing anything which you do not fully understand or in which you admit any liability for infringement.