It would make more sense as "go in the mud on a bike" but once you try to force it to this order that necessitates at the very least commas: ". . . never go, on a bike, into the mud." You're safer sticking to "ride" with your limited English . . .
I put "Arthur would never go biking in the mud" and it was not accepted. I reported it. I originally thought "Arthur would never bike in the mud" but thought that was perhaps a bit sloppy and might not be accepted. I thought "go biking" should be.