I will throw this out there but it would be fairly invasive but should be very clean once finished. It would consume the least space underneath the bed:

Four large acme lead screw pillars that the bed frame rides on. You probably want those lead screws to run all the way to mountings on the ceiling for stability and safety unless these lead-screws are unreasonably.

You will still have the issue that the if driven independently the motors will not synch up without electronic controls so you can choose whether to install one or whether to gang up the four lead-screws and drive them with a single large motor.

You will need brief access to a lathe at some point to turn the ends of the acme screws down so it can free-rotate in bearing mounts at both ends.

I will throw this out there but it should be the most compact but is also probably the most invasive. It would consume the least space underneath the bed:

Four large acme lead screw pillars that the bed frame rides on. You probably want those lead screws to run all the way to mountings on the ceiling for stability and safety unless these lead-screws are unreasonably.

You will still have the issue that the if driven independently the motors will not synch up without electronic controls so you can choose whether to install one or whether to gang up the four lead-screws and drive them with a single large motor.

You will need brief access to a lathe at some point to turn the ends of the acme screws smooth and provide a shoulder so it can rest inside bearing mounts at both ends, and also to mount a gear or sprocket so they can be driven.

Using a high thread pitch acme screw will allow the use of less powerful motors.

I will throw this out there but it would be fairly invasive but should be very clean once finished. It would consume the least space underneath the bed:

Four large acme lead screw pillars that the bed frame rides on. You probably want those lead screws to run all the way to mountings on the ceiling for stability and safety unless these lead-screws are unreasonably.

You will still have the issue that the if driven independently the motors will not synch up without electronic controls so you can choose whether to install one or whether to gang up the four lead-screws and drive them with a single large motor.

I will throw this out there but it would be fairly invasive but should be very clean once finished. It would consume the least space underneath the bed:

Four large acme lead screw pillars that the bed frame rides on. You probably want those lead screws to run all the way to mountings on the ceiling for stability and safety unless these lead-screws are unreasonably.

You will still have the issue that the if driven independently the motors will not synch up without electronic controls so you can choose whether to install one or whether to gang up the four lead-screws and drive them with a single large motor.

You will need brief access to a lathe at some point to turn the ends of the acme screws down so it can free-rotate in bearing mounts at both ends.