I set the M208 Z-1
My mistake. I've checked with @dc42, and this isn't necessary. Bed mesh compensation is applied after the limits have been checked, so the nozzle is capable of going below Z0 when bed mesh compensation is enabled.
Are you sure bed mesh compensation was enabled? Was the first layer exactly as it was before?
Have you confirmed with a straight edge that your bed is actually the same shape as the mesh is showing? It is still possible this is a measurement error of the Z probe; for example if the weight of the extruder is causing the X axis to deflect, and either bend in the middle or rotate around the X axis. That would also give a bed mesh of this shape, though it would be an extreme example.
Perhaps you could post a picture of the X axis setup of your printer.
Another option, though slightly tedious, is to do a manual bed mesh. Set the probe to type 0, and choose a 4x4 bed mesh, then run a new bed mesh (save the old one with a name first, with G29 S3 P"mesh_name.csv"). This will probe the bed, but for each point it will ask you to jog the nozzle down until it touches the bed (or grips a piece of paper). This should help confirm if the bed shape is the same or different from what the Z probe sees. Something like:M561 ; clear any bed transform G28 ; home as normal M558 K1 P0 H5 F300 T8000 ; create a new manual probe M557 X25:275 Y90:320 P4 ; define a 4x4 bed mesh G29 K1 S0 ; probe the bed with the manual probe and enable compensation G28 ; home to adjust bed mesh to Z0