Firstly have the supermarket written to say that they have accepted liability or is it a without prejudice offer? They could have investigated the incident and found they have a solid defence, but in order to save further costs in defending a claim, they have made a low goodwill gesture to settle as to save costs later.
I work in insurance and defend liability claims. From my experience, the big firms of solicitors are not that great and over 50% of claims will be handled terribly. They tend to use general template letters, but do not edit them in anyway to fit the individual claim, so often they are sending letters that have nothing to do with the incident circumstance. They also get things wrong that can reflect badly on the claimant (for example mr x was unemployed at time of incident but later started working, solicitors have their facts wrong and keep saying claimant was working at the time and want to claim loss of earning for the period, despite claimant telling us that they did not work at the time so this does not apply). They also very regularly will bring the claim against the wrong party, or refer to the defendant incorrectly (had one claim in which they kept referring to the defendant as McDonalds, we have nothing to do with McDonald’s at all). The last claim we defended at court, the solicitors had handled it so badly, misadvised the claimants and messed up sending in documents such as witness statements and a court bundle, that the judge actually advised the claimants that they should look to seek damages from their solicitors.
In addition, when injury claims are settled, they are not looked at any differently in regards to offer made if there is legal representation or not. The same calculations will be used to take into account details of the injury and any medical history that could be contributing (for example, someone claiming a back injury but who has a history of back problems would likely get a reduced offer).
You need to think about whether pushing the issue is in the best interest for this elderly relative too, you said that claiming wasn’t their idea and it was family members that have pushed this issue. By pushing further, you risk getting no settlement at all and you guarantee prolonging any stress for your relative.
If you do not agree with the offer, why don’t you write and ask how they have come to that figure, and advise of what you think of what region it should be and why you think that. Also ask if they have admitted a breach of liability or if this offer is made on a without prejudice basis.