A Wizard's Lizard
Binding of Isaac, is that you? What do you mean "no"? You sport the same visual style and gameplay. Hell, you also demand that a playthrough be completed in just one run, without saving or suspending your game. Oh, you are actually not grotesque... and you can also be played with a gamepad or a mouse. Cool, cool.
As you surmised, A Wizard's Lizard is like Binding of Isaac; a roguelite top-down shooter with room layouts similar to the dungeons of the first Legend of Zelda (unlike Isaac and Zelda, though, the game has larger rooms that span more than one screen). It doesn't have the variety, nice soundtrack, length or the grotesque content and it's also pretty easy, thanks to the following factors:
1) Gold. It's plentiful and you'll end up having far more than what it's necessary to buy anything expensive that the shopkeepers might sell you. Yes, that includes health items.
2) Even if you die, it's not the end. You'll respawn at the spot as a ghost and you'll have to fight the spirits of the dead enemies, as well as those you previously encountered. If you die again, then
it's game over. If, on the other hand, you find the special room with the red symbol (there is 1 per area), then you'll return to life with half of your health. If you then die again, you'll turn into a ghost again and, etc, etc.
3) Aside gold, there are plenty of drops, like health items and soul orbs. The latter one can be used to deal even more damage to multiple enemies and be further enhanced by the availability of multiple equipment parts (head, hand, belt, boots, etc).
In the end, if you are looking for a relatively easy and short roguelite with a bit of variety, A Wizard's Lizard
might do the trick. Otherwise, I wouldn't recommend it. Full list
Link to the official site (since muntdefems asked): http://awizardslizard.gamepedia.com/A_Wizard%27s_Lizard_Wiki