Yeah, more lives would mitigate the issue somewhat. I think giving player more or less lives can even be done independent of other changes. For example, it could be tied to a difficulty setting (Normal: 3 lives, Easy: 5 Hard: 2, Expert: 1 etc.).
I like the idea of lives sending you back to the physical mode the most, but you could also handle it some other way.
For instance, if you want to keep lives sending you back to a previous level when you run out, you could group levels into chapters. Every couple of levels you go to the next "chapter" (or whatever you choose to call it), dying could still send you back a level within a chapter, but it wouldn't send you back into previous chapter. It'd give some sense of permanent forward progression to the player.