Arduino + L9637 K-Line reader - 10400 baud on SoftwareSerial?
I've found 9600 baud to be the receive limit among standard baud rates (transmission can go faster). You're proposing an 8% increase over 9600 which sounds like it has a good chance of working but you'll need to experiment to find out.

You also asked why might your data (on the LCD) be wrong.

You'll do your sanity a big favor and do two separate experiments:


First, get known data (literal strings or terminal input) to display correctly on your LCD, before you try to use the LCD to confirm or debug your communications.
Second, get SoftwareSerial running (or find out that it doesn't...) at 10400 baud. In fact try to get up to, say, 12000 baud, or at least push the rate to find out where it starts to fail so you know how much head-room you'll have at 10,400. The simplest way would be to use two Arduinos if you have access to another one. Again, use your terminal or some literal strings on MCU 'A' to send to MCU 'B', MCU 'B' should receive a line and echo it back to 'A', who prints it on its terminal. Printing it on 'B's terminal wouldn't be a bad idea either.


You'll be in a lot better position to combine the two techniques once you have confidence that each one works; especially if you copy-and-paste known-working code segments.