This has been effectively solved by three different tools. Take your pick from:
Naturally they all have their own upsides and downsides, but if the only thing holding you back from shipping to linux is concerns about fiddling with shared libraries, just pick one at random and move on. I suggest appimages as feeling closest to a fat binary without triggering various licence clauses, deliver it like it's an exe.
I've built longer lasting pure linux binaries that use SDL with just gcc myself