I spent hours doing attempts and finally it clicked. If you use the OOB in Caves you can not only get over the wall glitch but also possibly make it faster.

After the last record, we decided it was best to only allow skips that unlock the next area, otherwise it wouldn't work with a fresh save file. This skip only has a chance at loading so I will allow it but only on runs that start with a new file and show stage select at the beginning of the run, so I can verify that it unlocked during their run. This will only work as a back-up method with the current set-up, but the option is still there if anyone does a run and wants to test their luck, if caves doesn't load.