When a prospective client invites himself to the first summer party at the law firm executive?s house, the executive must make sure the client is satisfied so he is won over while also dealing with new crises that arise in his home life.

It is a bit long. You can tighten it up by being more concrete on the goal and obstacles. (A collision between family and professional life always occurs at company parties, so this bit does not add much to the logline.)

Don’t know if this fits your story, but something like this might:

An executive hoping for a promotion organizes a company party, but when an eccentric client shows up, [something happens] and the executive must [do something] to save the party and his career.