The way I would think to do it is to write a large proposal, going into depth how the current solution is bad and how it should be improved.
But of course this is also a lot of unpaid work, preferably I would like to get paid for it, but my experience is in having companies ask me to build things, not suggesting to companies what I should build for them.
The company under question is also very compartmentalized/arranged under sort of feudal lines, I have some connections in the company but not in the area that needs to be improved so I wonder if my best entry should be to contact someone at the top of the organization and make my pitch.
Suggestions on how to proceed appreciated.
Try to convince the people that have this power, and then leave it up to the organization. With the best intentions, some companies will not change due to ignorance, fear, resources or culture.