In Umklapp scattering, the crystal lattice gains the negative of the reciprocal lattice added during the scattering event. So the "true momentum" is conserved. entire crystal due to phonon scattering is generally too small to measure, anyway.

Does phonon keep momentum?

It is true that phonons/sound waves don't carry net physical momentum (as long as relativistic and non-linear effects can be ignored). Phonons are merely collective excitations in the lattice. Since the time averaged momentum of each atom is 0, the total momentum of a phonon is also 0.

What is the significance of crystal momentum?

Physical significance

gives the state's periodicity, which is not the same as that of the lattice. This modulation contributes to the kinetic energy of the particle (whereas the modulation is entirely responsible for the kinetic energy of a free particle).