I claim the ornamental design for a lollipop support apparatus with base, as shown and described.
FIG. 1 is a front elevation view of a lollipop support apparatus with base;
FIG. 2 is a side elevation view thereof;
FIG. 3 is a top planar view thereof;
FIG. 4 is a bottom planar view thereof;
FIG. 5 is a side elevation view thereof shown in environment; and,
FIG. 6 is a front elevation view thereof shown in environment.
The broken lines of the base and header of the device showing in FIGS. 1-6 and lollipops in FIGS. 5 and 6 are for the purpose of illustrating features of the device and form no part of the claimed design.