The centrifugal field due to the rotation of the earth
plays no role in producing the tides. It gives rise to
the ellipsoidal (non spherical) shape of the earth, but
that is a background constant. The tides exist relative
to that.
Therefore you can save yourself a lot of trouble by
analyzing the case of a non-spinning earth. The tides
will be the same. Not spinning once a day, not even
spinning once a month or once a year, but fixed in
orientation. The stars never rise or set.
There remains however the idea of orbiting without
spin. The earth orbits around the center of mass.
The result is sloshing, as when you slosh a bowl
of water, moving it around in a circle *without*
spinning it. This gives rise to something you may
not be familiar with. It is sorta like the usual
centrifugal field, but not quite. I call it the
sloffugal field ... a portmanteau of "slosh" and
"centrifugal".
At any moment, the sloffugal field is the /same/
everywhere in the sloshing reference frame ...
everywhere within, on, and above the surface of
the earth. This stands in contrast to an ordinary
centrifugal field, which is proportional to r.
For details including diagrams, see the references
cited above.
The sloffugal field is just big enough to hold the
earth in its orbit, counteracting the barogenic
gravitational interaction. It contributes precisely
nothing to the net tidal stress.