I am a little nervous in answering this question since I have not seen any specific work on this subject and have not tried any calculations. So this is my educated opinion and not a definitive answer based on direct research. Short answer is no. The moon is too far and small (and unmagnetised) to effect the solarwind flow at Earth. Large space weather events would obviously have to pass the moon to reach Earth, but only a tiny part of the storm. HOWEVER - during the total solar eclipse the Earth's ionosphere does respond to the sudden lack of EUV radiation from the Sun. This was known and studied decades ago.
The solar eclipse effects the earth's ionosphere at short time scales i.e. during the period of the eclipse. Normally, the ionization drops during the time of the eclipse as the source is cut off and because the shadow moves at supersonic speed, perturbations in the earth atmosphere system occur. These perturbations may propagate from the troposphere to the ionosphere get amplified and seen as modulations in the F region ionization or total electron content. Many reports on the effect of total solar eclipse on the ionosphere/ atmosphere are available in literature.
Regarding the lunar eclipse, since the eclipse occurs during fool moon, there may be corresponding effect on the oceanic tides.
During my university studies, I had carried out some simple experiment. Attached please find one article. It may be useful to you. I know it was TOO BASIC in nature, but we really got some interesting results out of it. Hope this helps you.