I have quite a bit of experience of these types of situations.
Firstly you absolutely do not have to provide Neighbour A with any recordings or information. The meeting was correctly held behind closed doors. Even if a FOI request was submitted you can simply reject it for the reasons above. The only action open to Neighbour A is to take the council to court, but in order to do so they must have a case to argue and the onus is on them to prove there’s a problem. If this doesn’t happen the case is unlikely to be heard.
In relation to what do going forward is a little muddled, here’s why.
You’ve voted at a full council meeting to accept the offer from Neighbour B. This means the vote is legally binding. The only way to change this is for 2 or more Cllr’s to call an extraordinary meeting or wait for the next full council meeting and call the vote back in. There should be clear justification for this though.
The council should also be aware that it should be obtaining the best price possible for any assets and services. Therefore the higher offer from Neighbour A should perhaps be considered and form the opinion whether or not you wish to call the previous vote back in to.
However if you have had the land appropriately valued and you have achieved at least the value of the land from the current offer then that is justifiable, you are not obliged to accept a higher offer.
Essentially though until any contract is signed you are permitted (subject to the council retracting the vote) to accept some gazumping someone else. For anyone who’s gone through gazumping it’s a heart ripping process and can be seen as brutally unfair but it is not illegal.