What bit would you recommend as the next step?

Waterfords are good for pullers. You might also like a twisted wire snaffle or a slow twist. I had a big TB mare that used a thick eggbutt at home but out at an X-country event she had to have twisted wire. She was a LOT of horse and I weighed like 110 lbs back then LOL.

You're going to hear "school school school" and "circles" and yes that may be true.

But stay away from a Pelham. That's like 10 times harder than your eggbutt.

Try a thin eggbutt snaffle but before going to a really harsh bit see if your riding position is correct also have her teeth checked. I don't think a french link will help much you could try a Dutch gag or even a bit with rollers on it. Also use a flash nose band or a drop nose band so your horses dosen't open it's mouth to avoid the pressure of the bit.

Try a flash befor you step up to a different bit. It stops the opining there mouth. If they can open there mouth they can avoid pressure from the bit.

If she is too forward I would try a d ring snaffle nothing to harsh or she could et frustrated I moved my horse from an egg butt to a d ring and it made a big difference but I would still consult your instructor

Your horse doesn't need a stronger bit. What your horse needs is to be taught to respond to the bit you're using.