> then wouldn't that require a razor's edge of criticality?

Yup. From the device description in its decommissioning plan:

  The CFX was a sub-critical assembly of uranium-2-35 surrounding a Cf-252 source. The function of the U-235 fuel was to multiply the neutrons coming from the Cf-252 source, which fissions spontaneously. The CFX was designed never to exceed a Keff of 0.99. The CFX assembly yielded sufficient neutron fluxes for applications such as neutron activation analysis.

Keff is the fission neutron multiplication ratio; 1 is criticality.