If you have a solicitor on board you might consider a “round table” meeting with lawyers. Sometimes this is physically around a table but more often it takes the form of a “shuttle” type negotiation.
The lawyers would shuttle between two rooms where the clients are. That way the clients do not have to see each other, and they have the advice available to them.
This sort of negotiation can also be practiced with a mediator although usually mediation does not take place with lawyers physically present.
Many solicitors will also offer what is known as collaborative law. This is where the clients sign an agreement at the outset of the process by which they commit that they will not use these lawyers to litigate the matter should they fail to reach agreement. This is an incentive on the parties to remain engaged with the negotiation process for fear of incurring the expense and inconvenience of having to instruct a different lawyer who may not be their first choice. The meetings are conducted around a table with both lawyers present and work in the best interests of both clients.