That neck joint really doesn't look good at all.
The neck inlay seems to be from an ES-3 (not currently for sale on the site), not the top diamond tooth inlay the GR-1SF should have. Here's the neck and body from my ES-3 when it arrived. Apart from the cutaway style and a few hole locations. they are very similar kits.
The GR-1SF neck's headstock should come with the two rounded 'scrolls' on the end, as in this PBG website picture of the kit.
If you haven't got that, then you may well have been sent the wrong neck for the body. If so, it looks like you may well need a new kit.
Whilst you have a lot of side area for gluing on the neck, the bottom the heel to the body joint is the most important one as that's the only part you can really clamp together for a strong joint. With that big a gap between the heel and the body, I wonder if any of that heel is touching the bottom of the pocket?
I'd also be looking at checking the neck angle,as it seems if you get the neck sitting flush to the top, then you get a gap between the heel and the side of the body. If you have it so there's no gap between the heel and the body, you'll end up with a gap between the neck and the top of the body. Which means that the neck angle can vary significantly. On my (and others) ES-3, the neck angle is too shallow and I cant get the bridge low enough for a good action. I've replaced the bridge with a lower one and tried filing things away, but still no luck.
So I'd straighten the neck so it's flat, run a straight edge along it and see where the end of the straight edge ends up where the bridge will sit. See how much of a gap you've got there and will the bridge fit in at its lowest setting (allowing for post insert rims etc)? If the bridge does see-saw, then does the neck need to be pointing up or down in order to get the bridge in the space under the ruler's edge.
If the angle looks OK, then you could glue layers of veneer to the bottom of the pocket to fill in any gap between the heel and the pocket bottom. I'd shine a light through the gap to see if the gap goes all the way through or not.
I tend to fill small cracks/gaps with a filler I've tinted with artists acrylics to to match the finish. But the current gap by the heel cap is pretty large, so if that still exists (or it's filled with a couple of layers of veneer that look ugly), then other people have added an extra block of wood as an extra heel cap to cover the join. It could be basswood to match the body, or it could be a dark wood to provide a contrast.