There is high demand for a client only version, and it's theoretically possible to do so by checking for server entity motion packets.

However, I believe the health changing packet is sent before the motion packet, and this causes an issue with 1.14+ (and perhaps older versions that are lagging?), where the camera starts to tilt before the motion packet is received, causing an ugly snapping effect.

I'm trying to theorize possible ways around this. Perhaps a solution would be to delay the screen shake until the next motion packet is received? Any suggestions or help is appreciated.

Assuming the motion packets you're referring to are those for the player's knockback, I'd just delay the tilt until the knockback is recieved. It shouldn't really cause problems unless you have a lot of lag (and is also slightly more realistic, since the knockback is what would actually cause your head to tilt).