So am I reading that scope correctly in that there is continuously over 1v potential difference between the ground at the Arduino and the ground on the LCD module?
That looks like a ground loop issue or poor grounding vs ground bounce.
That would imply something very serious somewhere on the PCB. Either bad etch or a short somewhere.
With ground bounce, the signal will typically be at or near ground most of the time, but then under certain situations, typically when some sort of signal switching occurs, the signal will be briefly yanked up 1 or more volts, and then will fall back down.
Does the ground signal stabilize back down to to ground if you hold down the reset button on the Arduino?
And maybe you have both, where the signal isn't properly grounded and then experiences bouncing.
Some decoupling can sometimes help against bouncing.Like adding a .1uf up by or on the actual LCD PCB.
The issue with ground bouncing is it can and often does occur when the host (arduino) is sending a nibble, so what can happen is the first nibble is sent, which causes a bounce and then things go wacky on the next nibble because the signal levels will appear unstable to the LCD since they will be right at the threshold of high vs low.
This could cause the LCD to not see the next E signal transition, or to see multiple E transitions when there is really only a single one, or to misinterpret bits within the nibble.
All of those are bad and can cause the LCD to lose nibble sync with the host.
And once that happens, the LCD will start to behave in unpredictable ways as it will not be receiving garbage data/instructions.
If you are thinking about switching to i2c for the LCD and it can work for your design, I'd go ahead and do it.
The wiring (and PCB) is much simpler and you can use something simple like a 4 pin header on your PCB to connect the 4 LCD wires to the 4 pin header on the backpack using off the shelf 4 pin header wiring.
You could accommodate both. Put the traces & holes for using arduino pin control and 4 holes for using a 4 pin header to accommodate an i2c header.
The most common i2c header wiring is SCL,SDA,VCC,GND
--- bill