If I attack with Glasc Mixologist into a battlefield my opponent controls, and all creatures die, can I play the unit from the Deathknell to that contested field?

No, you cannot play the unit from your Deathknell ability to that battlefield.

The Ruling

According to FAQ #9314, when you move a unit to a battlefield controlled by your opponent to attempt to conquer it, you are the attacker and you do not control that battlefield. Because you do not control the battlefield, you lack the permission to play units there.

Sequence of Events

1. Combat Damage: You attack with Glasc Mixologist into an opponent-controlled battlefield. Damage is dealt, and Glasc Mixologist dies.
2. Deathknell Trigger: Because Glasc Mixologist has a Deathknell ability, it is added to the chain as a Pending Item (Rule 428.1.a.1.b and Rule 323.4).
3. Resolution Attempt: When you attempt to resolve the Deathknell ability, you are instructed to play a unit from your trash. Because you do not control the battlefield you are currently in, you cannot play a unit there.

Why this is the case

As the attacker, you only establish control of a battlefield after the showdown is resolved and you have successfully conquered it (Rule 464.1). Since the combat is still ongoing (or in the cleanup phase immediately following combat damage), you have not yet established control, and therefore cannot play units to that location. This differs from a defending player, who already controls the battlefield where the combat is taking place and is thus permitted to play units there via Deathknell (FAQ #9149).