I have asked a similar question to this previously
Communicate with Cash Drawer from Website
However the question was very vague and I now know what I requir
You are going to need to write a Chrome extension that exposes an API to your website. This is the easy part. You inject a content script to your website, and then use chrome.extension.sendMessage to communicate back to your extension.
The hard part is to actually open the bat file from your extension. With NPAPI, this would have been easy, since you could just write a C++ plugin that calls CreateProcess or something. Contrary to what you said, it's a pretty good solution. But unfortunately, NPAPI isn't an option, since it's being deprecated.
So what you should do is to use Native Messaging. Basically, it's a Chrome API that allows extensions to exchange messages with native applications using standard input and output streams.
Read more about it here: https://developer.chrome.com/extensions/messaging#native-messaging-host