This year there is no Stuff a Stocking achievement, so, is it impossible for me to gain Melody's Little Helper achievement or are there alternative ways?

Stuff a Stocking achievement is retired. So I'm pretty sure you don't need it for the Melody's Little Helper achievement. I have the Little Helper achievement and it says I'm missing Melody Regular so I think that replaced it.

Yiu are right, today I'm missing only Melody Regular. Thank you :-)