>>>"they have asked me for more, which I have verbally agreed to"<<<
AAAGH!!! No! No! No! A thousand times no! (Er, except as stated below, of course!).
As executor of the will, the law obliges you to pay out exactly what the will states, not a penny more nor a penny less. Any other action is ILLEGAL!
However, of course, you're free to give your OWN money (including what you're due to receive from the will) to whoever you want to. If, by giving more money to the 'awkward relative' than the will states, you're simply depriving yourself (and no other beneficiaries) of what you might receive, that's entirely up to you.
>>>"Solicitor advised me against this . . ."<<<
Well, at least the solicitor seems to be earning a bit of his/her money anyway!
but
>>>" . . .and said they should get their solicitor to contact him to negotiate"<<<
Just how badly was the will drawn up? A will normally leaves either a specific legacy (e.g. "£1000") or a specified fraction of the residual estate (e.g. "five per cent") to a beneficiary. Either way, simple mathematics should dictate the sum to be paid, not negotiations between solicitors!
Read the will again. Then read it once more. If you're absolutely convinced that you've assessed the correct sum to pay to the awkward relative, pay that exact amount (to the penny). Unless you're expecting that someone will suddenly prove that the deceased person owed them thousands of pounds, you can probably ignore the 6 month period referred to in your post. It certainly has no bearing upon any potential claim from any beneficiaries of the will.
Chris