I'm not an engineer, but is there a reason why they can't french press (sans filter) the fuel toward the bottom of the tank as it empties?
Instead, it's more common to use gasses injected at the top of the tank to push the liquid to the bottom. Falcon 9 uses helium. Starship uses https://en.wikipedia.org/wiki/Autogenous_pressurization as well as small header tanks for the landing propellants.