Isn't this no different to a First employee stopping you reach the coach terminal for a Stagecoach bus? It's nothing to do with the company operating the barrier, and to prevent someone who is NOT their customer, from using a perfectly valid ticket on a rival operator and attempting to extract additional money from that customer is totally unacceptable. It would not be accepted in ANY other industry. So why are they allowed to get away with it on the railways?
I'd write to Virgin asking what steps they will take to ensure LM do not act this way again.
I'd also write to LM stating that you were a customer of Virgin's, not theirs, and their staff had no right to obstruct you. I would inform LM that such actions are less likely to make me choose to use LM tickets in future, due to the behaviour of their staff. I would ask what steps they will take to ensure that staff look up the relevant restrictions, not either make them up or try to guess them!
If no satisfactory response, write to your MP stating that you are disappointed at the lack of integrated public transport, and is the Governments idea of integrated transport that a representative of a company who is nothing to do with the company you are travelling with is able to prevent you travelling on your booked service, and what are they going to do about it?
Don't accept being fobbed off by either company.
Be polite, but firm and assertive in the letter you write. You are welcome to PM them to me for proof reading if you like, I will be happy to assist.
Although the LM employees at MKC clearly acted incorrectly, there is an excellent member of LM staff on here who may be able to assist....