I still take the view that in business circles in the UK this is not sending out a mixed message. The connection on LinkedIn would have had to be either naive or deliberately obtuse to think of it as a social invitation. He was from a local firm of solicitors and they should be training their staff better on how to do business networking. If the OP or any of her colleagues have other connections with the firm it might be as we'll to drop a word in the ear of one of the partners. I know we'd be appalled if one of our staff did this as it could affect the reputation of the firm.