Spiderhax, otherwise known as WebKit exploiting, is possible without the game. It is used in Yifan Lu's spider3DStools to gain ARM11 kernel access, and I believe it was used in another exploit system to gain ARM9. It crashes the browser into a panic while injecting a binary. Yifan Lu is already working on a way to open the Homebrew Launcher's boot.3DSX via the browser in his exploit dubbed spiderninja, which you can see below, and was only shown for a bit before removed. In the Regionthree Readme, Smea states that the exploit will not work with SKATER, but the code is there, and he hinted that anyone can do it. I haven't looked much into SKATER, as I do not own an N3DS, but it's pretty safe to assume that it still uses WebKit, meaning it should still be possible, with a bit of rewriting. You seem to be very dependent on a game, which ultimately, is not something a long term exploit should depend on.
Why wouldn't it be? If SKATER is still relying on WebKit, which I am pretty sure it is, I don't see why it wouldn't work on it. Again, I'm just working off of what Smea hinted at in his Readme.