Should I try switching the cable or is the latency more associated with the iPhone/monitor (which I can't replace)?
You could try switching out the cable, but you’re not likely to see much improvement. Your device itself isn’t the problem here, it’s the conversion from USB to a digital video signal that’s the culprit.
It’s technically a USB to HDMI adapter
The Lightning port does not supply a native video signal; just four data lines. As such, it only supports a small number of signals: USB, charging, and audio (via a DAC). This means that the video must be “created” to connect to your display/TV.
Basically, there is a typical USB to HDMI (or VGA for analog) adapter embedded in the cable; it’s actually just the chip. That chip is just enclosed in an “all-in-one” cable that goes from the Lightning connector to the HDMI connector.
Creating video takes time
This video signal creation means there is an inherent latency no matter what you do. It takes time to create that signal and it will be nowhere as fast as the embedded GPU in your iPhone.
Get a better cable…
You could potentially reduce the latency with a higher quality cable from a reputable cable manufacturer(I am partial to Anker, CableMatters, Belkin) or get the Apple Lightning to Digital AV adapter. (IMO) The cable that you linked is from an unknown brand (likely Chineseium); it’s a red flag when they pepper their Amazon title with every potential keyword like they did.
TL;DR
Bottom line, try a better quality cable to reduce the latency, but in this case, you cannot eliminate it. It’s also important to remember, this feature was meant for mirroring your iPhone display for convenience, not to be an external display in which performance is a factor.
