Maybe the Botany Bay
just had an oddly shaped centrifuge interior and artificial gravity was created the old fashioned 2001 or B5 style?
SCOTT: Well, it looks like that ship was expecting us. We read heat coming on, complete oxygen atmosphere.
Considering how the TOS producers emphasized the vital role of artificial gravity in the Writer's Guide
and especially its limited availability while on battery power only ("In case of total failure of all engine power sources, the vessel's gravitational and life support systems can be switched to battery power, with a full-load capacity of about one week."), add to this that the Botany Bay
apparently only had nuclear power (the equivalent of a 23rd starship battery?), it looks like it had to rely on centrifugal AG.