Why do you want to use G-sync in the first place (if you don't mind me asking)?
RE G-sync over HDMI; I would recommend not even trying. With the right card (you will need HMDI 2.1, which you are unlikely to have access to) and the right monitor, it can be done, but save yourself a lot of trouble and do it via Display Port OR - and I only figured this out recently - it can be done via USB-C (assuming you have this capability on both ends of the chain, which is of course not guaranteed).
As for the 3050, the recommendation to at least get a 3060 is good advice - I have a 3060 in my laptop, and it's perfectly serviceable, but I'd definitely have gotten something a bit beefier (especially in a laptop, since the GPU can't be upgraded).