The tailpeice is adjustable for this exact reason.
I topwrap my guitars but it's not really necessary You cna just raise the tailpiece. The OP's tailpiece hieght looks perfectly fine. Nothing too extreme there.
Buy a dual P90 SG and swap out either or both of the P90s for soapbar size humbuckers. There are a lot of options available today.
Here's my Les Paul Special with Dimarzio soapbar humbuckers. There were a direct drop-in replacement for the stock P90s.
I have a 1986 SG62. One of the tone pots started failing so I replaced the all the pots with 500K audio taper. Made a huge difference. Brightened up the tone and gave it a much more "open" sound to it.
Has a much more useable range on the tone pots now. I can back off on them if I want it...
I had Dimarzio 36th Anniversaries in my SG. The neck sounds fantastic but the Bridge sounded a little thin with the amp I was using at the time. I put in a Duncan JB at the suggestion of another member here and Love it.
Showing a frontal shot of the 86 vs the other two might help explain what the OP is experiencing. The 86 should have the fretboard extending onto the body a bit more posiibly giving it a shorter feeling neck.
Here is a pic of the neck angle. These are not SGs but it will give you the idea.
On the top pic you can see that the neck is at an angle in comparison to the body. The bottom pic shows the neck with no angle. It is parallel with the body. Yours is probably like the bottom pic and your newer 61...
Take a close look at the line under the tuner. It's either a finish crack or a crack in the wood. If it's a crack in the wood I would take it back to your local guitar center and see if you can get some money back.