Would a satellite in geosynchronous orbit between the earth and moon track across the sky together?
The answer to the headline question is clearly: No.

The full moon has an apparent diameter of about 0.5°. Its apparent motion is roughly 360°/24h, neglecting the orbital motion of Earth and Moon. Hence it apparently moves about 24h*0.5°/360° = 0.033h = 2 min.
Hence the moon moves one moon diameter in about 2 min.
(The neglected orbital motion induces an error of about 3.65% (0.997d/27.3d), divide the siderial Earth day by a siderial moon orbit, hence an error of just a few seconds for the apparent motion of 0.5°.)
In 35 minutes, Moon's apparent motion would be about 17 moon diameters.

The geosynchronous satellite, in contrast, remains staying at the same apparent position, by definition.

Planets or stars move apparently about the above error estimate different from the moon, hence roughly 27.3-times slower relative to the apparent moon than a geosynchronous satellite.