Hm ... that's really interesting. I'm reading about it on NesDev. Is there any specific reason why this occurs?
Splitting the screen to have a stationary status bar with a scrolling main display requires a timed write to occur in the middle of frame rendering.
Neither SMB's cartridge nor the NES itself offers an IRQ which will cut in when the appropriate time occurs, so the game uses sprite 0 hit to mark it.
If you change the sprite, and not the matching BG tile (or vice versa), it changes when the "hit" occurs, and thus screws up the timing of when the game splits the screen.