How can I turn a MOSFET into a diode?
Enh. FETs have a (Vgs-Vt)^2/Ids impedance while diodes have a log Vf/If slope with a bulk Ri at rated current that approximates towards a linear curve above Vf@If max.

Apparently FETs made with this desired sub-threshold characteristic will behave like diodes in the low current region below gate threshold.

But you can't bias just any Enh. mode FET to behave this way. The log slope can be made to match a given diode characteristic by design, but that design is beyond the scope of this question which gets into "weak-inversion current reduction"