Strain of Ash and Mipedian Balladeers' Flute

Strain of Ash reads, "Sacrifice a Battlegear equipped to an unengaged UnderWorld Creature you control. If you do, destroy target Battlegear."

Mipedian Balladeers' Flute reads, "Sacrifice Mipedian Balladeers' Flute: Equipped Creature gains a Mugic counter. Your opponent may put a Mugic counter on a Creature of their choice."

If I use Strain of Ash and target this battlegear as my sacrifice, will I be able to still obtain the mugic (allowing the sacrifice to have two effects) or will my Mipedian Balladeers' Flute be rendered useless.

I figured this wouldn't work on something like a Mugician's Lyre (face-down so effect isn't able to be triggered) but a Mipedian Balladeers' Flute is face up and they do both say "sacrifice" (not "destroy," for instance).

I hope this is clear. TLDR: Can both cards work simultaneously?

You cannot sacrifice the same for multiple effects. So no, if you sacrifice Flute in response to Ash, Ash's effect fizzles, it doesn't destroy anything and you lose it, but you're Creature gets a counter.

Ok that makes sense