Yeah, let's throw TM under the buss. The place is doomed. Maybe they do need to "try harder."
Look, I have been there and done this. We had less than 75% OTD to our largest customer. It was a complete cluster blank. They where looking to get rid of us. What OP needs is a structured approach to solving the issues, one bite at a time. They don't need "lean training." They need a little common sense and a unified goal. The goal has been set -- improve OTD. Now get into the details and go do it.
It is actually pretty simple. Track the shipments every day. What went out and was it on time or late. If it was late, determine why. Possible answers: capacity issues, not enough lead time, machine down, incoming material was late, tool needed repair, sat on someone's desk too long (i.e.; disorganized office), etc. Do it for a month, then pick the top 2 or 3 issues and make an improvement. Rinse and repeat. If you go to TM and say "the top reason we are late on jobs is this, and here are some solutions" they will work with you. If they don't, then you know.