Does the Earth ever cross the Moon's path with respect to the CMB?
Not within plus/minus a few hundred years at least! The Moon's orbital plane around the Earth (viewed in an inertial frame) rocks back and forth a bit (Lunar Nodal Precession with a period of about 18.6 years), but that's not going to help here. 

You can play with the numbers in this program.









import numpy as np
import matplotlib.pyplot as plt
from skyfield.api import Loader

degs = 180./np.pi

load = Loader('~/Documents/yournamehere/SkyData')
data = load('de405.bsp')
ts   = load.timescale()

timez = []
dayz  = np.arange(1, 31, 0.1)
dayz  = np.arange(1, 366, 0.1)

#for year in range(1700, 2200):
for year in range(2017, 2028):
#for year in range(2000, 2030):

    #time    = ts.utc(year, 7,  dayz)  # July
    time    = ts.utc(year, 1, dayz)   # January
    timez.append(time)

earth = data['earth']
moon  = data['moon']

RAdegs, Decdegs = [], []

for time in timez:

    RA, Dec, dist = earth.at(time).observe(moon).radec()
    RAdeg, Decdeg = degs*RA.radians, degs*Dec.radians

    # lazy way to block the wrapparound
    dRAdeg = RAdeg[1:] - RAdeg[:-1]
    dropit = np.abs(dRAdeg) &gt; 100
    RAdeg[:-1][dropit]  = np.nan
    Decdeg[:-1][dropit] = np.nan

    RAdegs.append(RAdeg)
    Decdegs.append(Decdeg)

RAdegpts  = [157.5, 157.5+180]  # coordinates from the question
Decdegpts = [-24.0, +24.0] 

if 1 == 1:
    plt.figure()
    for RAdeg, Decdeg in zip(RAdegs, Decdegs): #[::5]:  # plot every 5th year
        plt.plot(RAdeg, Decdeg)
    plt.scatter(RAdegpts, Decdegpts)
    plt.title("2017 throuth 2027 continuously", fontsize=20)
    plt.show()