Sounds like an “interesting” problem.
Powerline adapters are unlikely to work. I assume each floor has its own power metering? There is usually filtering in the power meters to attempt to reduce interference to the customer. Those adapters rarely work between separate metered circuits.
You might get a strong enough 2.4GHz WiFi signal between floors, but it is highly likely that your neighbors are using WiFi, and probably 2.4GHz. There are very few channels in that band, so interference is highly likely. 5GHz has a lot more space, so interference is less likely, but so is the prospect of that penetrating that much concrete.
One floor separation from multiple washing machines might not be as good as you would hope. You may actually be better off having the Shake in your apartment.
What I might do in your situation:
Read the documentation on setting up your Shake in “standalone” (no internet) mode, and run it for a few days in your apartment. Check the data to see what it looks like. Then repeat the experiment with it in your basement. Let the data quality guide you as to the best placement.
Assuming that the basement looks best, I would use a WiFi app on my cell phone to see what channels are in use, and how strong the signals are - can you see YOUR WiFi signal???
If you can see your own WiFi from the basement, even if it is very weak, then consider installing a point to point link using reasonably powerful radios with directional antennas. I use a pair of Ubiquiti Networks airMAX NanoStationlocoM 2.4 GHz.
Another option to explore, depending upon which country you are in, how old the house is, etc. would be to check for wiring conduits in the walls which probably run vertically to distribute (at least) power. Slipping a Cat 6e cable into there may or may not be easy.
If you can borrow a LAN over power setup, it may be worth a try. I wouldn’t buy one myself until I knew it worked, because it most likely won’t. It should not affect the Shake.