it's just my core not being strong enough for this weight

That's probably part of it, yeah. Try to focus on one smooth movement, not a two-phase lift like you're doing. Personally, I find your stance is a little wide for high bar. This encourages forward lean. If I was doing high bar I'd have a narrower stance and my toes would be pointed in a bit more. If ankle mobility becomes an issue, raise your heels on a board or small plates until you can gain additional flexibility. Just something to try.

Or you could try low bar, since you've already got the stance and hip drive going.