Using the cheat search in SNES9x 1.60, I was able to figure out a way to do a 'level select.'

When you restart or clear a stage, the game will put you on the level selected with the code; it's probably not easy to change this on the fly on console (I only have an Everdrive, you super cool people with the USB-enabled one might have more luck?), but on emulator you can simply update the value of the code to pick stages.

You'll need to know what number stage you need in Hex; 00 is the first stage, 5F is the room with the disco lamp before the Magikoopa at the end. 60+ are your individual Bowser Castle rooms (but 5E takes you to the first half of Bowser's Castle).

Code in SNES9X is: 7E13BF=XX, where XX is the level you need in Hex. It should work in other emulators if you format/convert it correctly; if you need help, leave a post or DM and I'll try to help.