Forget about it.. they will argue (probably correctly) that they had loaded out of that shipper previously and that they therefore had a relationship with that shipper prior to moving your load. It's a gray area. Also, back solicitation won't hold up in court if its too general.. your contract may restrict your carrier from hauling similar loads in the same lanes out of the same plant within a specific time frame; however, back solicitation clauses that are vague or too sweeping in scope carry no weight and are completely unenforceable. For example, if you haul a load out of GM for me I can't expect that you can never haul any more loads out of any GM plant ever again without incurring a penalty from me.
About lawyers.. don't do it. Flushing money down the nearest toilet would be easier and more productive. Most lawyers want a big retainer up front, and you will pay through the nose and get nothing back. In the vast majority of transactions that involve a lawyer there's really only one winner: the lawyer.
The best way to go is to recognise what happened and to discuss it with your customer. If the carrier did indeed back solicit you then your customer is an accomplice to that. Why did they allow that to happen? Talk to them to establish what services you can provide to make back solicitation a nonstarter for them. Ultimately we as brokers have to provide a value above and beyond what a carrier can provide directly.. if we can't do that then we become fair game for back solicitation. What can you do for your customer that a carrier cannot provide directly? Perhaps it's a diversity of services.. lots of different lanes, TL, LTL, rail.. etc.. Perhaps its volume... 50+ trucks a week.. Talk to your customer to discover what value you provide as a third party..