OP's mom wasn't IDB'd, she was rerouted (seemingly agreed to it voluntarily) and the agent marked it INVOL to streamline the compensation issue.

The mistake that the OP's mom made is mistakenly believing that MP and M&M are interchangeable programs and that one airline (UA) can credit good will miles to the other (LH). This is NOT possible and will not happen regardless of how hard the OP pushes. It simply can't be done.

My suggestion would be to reply saying "here's what was offered at the gate, my mom mistakenly requested mileage compensation instead without the understanding that UA could not credit to M&M, and the gate agent directed us to you for assistance. we would appreciate it if you would please provide the originally agreed compensation of $400." or something to that effect. still might not be at all effective, but probably the best you can do right now.