#1 would still be missing Art. Going from Dying to Time is a pretty big stretch. Lullaby to Art is pretty much the same, if i'm to make that kind of association i could just as well associate anything with anything. I could go from Evil to Death to make it fit with the Death rune on #3 for instance. Hell, i could even go from lullaby to death.
I don't know which was the intended way to solve this. Maybe all 4 runes were supposed to be spelled out in each text but the fact is that 'as is' they're not. Maybe the texts got screwed up somehow, maybe the runes did or maybe we're missing something obvious. Regardless, picking 3 obvious runes from each text and figuring out the positioning of the 4th by trial and error works, no wild associations required.
1. Most probably something was lost in translation. I guess for 2, 5, 6 the Art assumption works perfectly.
2. As for "quietly moans" (1), whether you argue it's Death, Art (or anything else) matters not, seeing that the position of Animal, Sky, and Time is clear.
3. For 3, "Dying on a fresco": The position of Animal and Sky is clear. If you have chosen Death for dying (which arguably have the same symbol as Art), this would conflict with "fresco" which is undeniably Art (the alternative, Time, is just too far-fetched). So there can only be one spot for the last rune.
4. For 4, "... of scarceness. An evil lullaby". Here the position Sky, Animal, and Time is clear. Whether you argue that the last one is Death (scarceness / evil lullaby) or Art (lullaby being a song so there is a loose connection), again there is only one spot for the last rune.
So I am arguing that even with the Death-Art ambiguity, you can still deduce the position of the final rune (Art / Death).
To go one step further - unnecessary but it is done for the sake of consistency (which might be moot given the doubts in the translations) - since point 3 has indicated that Death ("dying") is associated with Time, it invalidates the interchangeability between Death and Art, hence my opinion that in points 2 and 4, Art is the preferable interpretation.
Without going through this admittedly convoluted process, your suggestion of trial-and-error for the 4th rune makes perfect sense.