The gpi of the GT 340 trads is 12.1
You're not going to get a significantly flatter trajectory until you get the arrow weight down to 10 gpp or less, which in your case equates to an overall arrow weight of 520 grains or less (520 grains / 52# = 10 gpp). With that bow, you could safely shoot an arrow of 8 gpp or less, but for a hunting arrow I wouldn't go much less than 9 gpp, which would give you an arrow weight of 468 grains.
The .204 diameter arrow might give you a higher dynamic spine than a comparable .246 diameter arrow, since the .204 would be closer to centershot, but the difference wouldn't be very significant. Since the Satori has an adjustable centershot, whether you are using the included shelf rest or an attached rest, you might as well adjust the centershot to the ideal position, which is where the arrow tip is just outside the centerline of the bow.
3/8” nock height would generally be on the low side for a 3 under grip. If you were bare shaft testing, you could find the ideal nock height, but since you're not, I would set the nock height at 5/8”.