Why didn't my mustard sauce emulsify?
You really need to add the oil slowly at first, especially if whisking. You are trying to disperse the fat into tiny droplets suspended in the water phase (the water from the butter and the vinegar). You can't whisk fast or hard enough, especially by hand, to break up 3.7 parts fat in 1.3 parts water when all the fat is trying to combine into a single mass floating on the water, no matter how much mustard you have.

I would have whisked the mustard into the vinegar, then slowly added the butter, then the oil.