The ornamental design for a beveragecontainer handle, as shown and described.
FIG. 1 is a perspective view of a beverage container handle showing my new design;
FIG. 2 is a front elevation view thereof;
FIG. 3 is a top plan view thereof; and,
FIG. 4 is a side elevation view thereof.
The broken line showing of text on the beverage container handle is for the purpose of illustrating environmental structure that forms no part of the claimed design.