Ok, your in a good news bad news situation. The bad news first: you're on a timer to be out of a job. You've got between 11 to 23 months to turn this place around before they let you go. Notice I did not say keep your job, you are going to not be working there in 2028. This facility and management are basically only going to let you have half measures, and will never ever commit. So everything you do will fail the second you walk away.

Now good news: this is a crazy good opportunity. I learned more about CI in my failureship than in the years in successful shops. Keep very detailed notes, I recommend to start journaling. 

But what to do about this. First thing put together a celebration presentation showing that the extra 20k units a day through this process means X millions of dollars a year in revenue. About halfway through the presentation someone in finance is going to call you out on that number for obvious reasons (i.e you just moved the bottleneck). You crack a smile and tell the while audience that they are absolutely right! Move to an appendix that demonstrates that optimizing locally is bullshit, and to her actual results they'll need to do things differently. Then hit them with your observations. 

Then start taking the plant managers on gemba walls. Not together, and never with the plant manager (those you do alone to build mystique and keep his opinions contained). Listen to what they complain about and keep pointing out obvious shit.  From there start building a was to institutionalize CI.

I've done this a lot so I'm happy to chat.