I mean, the gps satellites themselves are also affected by the Moon's gravity. If the positions of the land and satellites of it are shifted roughly equally, I wouldn't see anything. Or if the satellites are shifted more (and roughly equally), I'm not really measuring land rise, but satellite rise over the ground.
Satellites can't be shifted as Earth's surface. Moon gravity simply makes their orbital plane slowly spin around some axis. But GPS supporting team constantly monitoring satellites motoin, Earth rotation (relative to quasars, btw!), and tons of varoius factors, so satellites orbits are defined with very high precision, and this data constantly updated. So GPG must get absolute error of few meters at any time.
The other thing, is increasing precision by making many observations and average them. The error will decrease as square root of number of observation. This is commonly used method in science. For example, single measurement of star velocity have precision of 50-100 m/s, but collecting data of many years of observation allowing to reach precision of few meters per second, and discover small exoplents orbiting it.
In your case, you must average only data obtained in the same condition. I.e. relative position of the Earth (including rotation angle), Moon and Sun must be equal. This is highly unpleasuble, because Earth rotation and orbital period is not related with Moon's orbital period, and Moon's orbit have precession with 18.2 years period. These strange secondary peaks may be explained by Sun's tidals, which is 1/2 magnitude of Moon's, and not synchronized with them.