Welcome to the forums.
I'm sure you understand that copyright infringement is a matter of strict liability and so your unintentional mistake would not in itself assist your defence in court if the matter ever went that far, which I doubt. However including it in your response does no harm as evidence of contrition. Whether PicRights take it into account is also something I doubt.
I assume that you haven't found any evidence to support your explanation of how you first found that image. That is a pity because if you were able to show how you were misled into thinking the image was free to use, that would certainly have improved your position. As a next best step it might be worth trying to find the same image being offered on a picture library website such as Shutterstock, and making a note of the cost the licence which was most appropriate for your use of the image. Chances are that the cost of a licence of that type will be in the vicinity of the figure you have come up with, but if you are able to find the exact figure, which represents the true market rate, then you will be demonstrating to PicRights that you understand your position legally speaking if they decide they want to take the matter further. Your draft response certainly contains what is necessary to constitute a counter-offer, but it might be improved by restucturing it slightly, so that your initial explanation is then followed by your remarks about not wishing to deny the photograper his or her due. Then follow up with a slightly stronger statement in support of why you feel that the amount of the counter-offer is reasonable. Being cynical, it is only this last part of your response which will actually be read by PicRights and they will be looking at its tone in order to judge how resolute you are and whether, ultimately, it's worth them pursuing the matter. They will of course follow their standard procedure, as you will have seen from the experience of other posters here, and tell you that they can't accept your offer and they will then propose a new figure, and so on. But there is a limiit to how long they will continue the chase before it is no longer economic to go further.
If they ultimately take you to court and you defend the claim*, you can expect the court to award damages which reflect the true market value (ie the cost of a licence from somewhere like Shutterstock) and you will also have to pay the court costs the copyright owner (presumably Reuters) incurred in bringing the case, namely the filing fee and hearing fee - around £150 in total - plus any reasonable travel costs and loss of earnings (not to exceed £90) for the claimant for the day of the hearing. So in all, quite possibly around the same as the £350 of their initial demand. However only the damages element will go back to PicRights and the copyright owner as a 'profit' and they will no doubt have incurred upwards of £1000 in legal costs which are not recoverable from you. From this you can see that this is not economically attractive to either PicRights or Reuters, which why I say I don't think that is a very likely outcome.
I hope this helps.
*I can't stress strongly enough that this outcome would depend on you robustly defending yourself and refuting the inflated claim of £350 or whatever figure they put on a court claim form. Failure to defend the claim would result in a default judgment and damages would be set at whatever figure the claimant dreamed up, since the court would not be provided with any counter argument as to why that figure was unreasonable.
Advice or comment provided here is not and does not purport to be legal advice as defined by s.12 of Legal Services Act 2007