Operation of BJT with low values of Vbe ("cutoff" region)
I understand your issue. Personally I would not expect either model (Ebers-Moll nor Gummel-Poon) to produce accurate results with such low currents. The models are neither designed nor verified for that region of operation.

Also I think that leakage and temperature related issues (and we all know that leakage and temperature have strong relations) will influence the currents dramatically. Often these effects are poorly modeled.

And even if the models could support accurate modeling at such low currents then the question remains if the model parameters are in any way reliable. If the transistor's manufacturer (foundry) did not measure and extract those parameters at such low currents then no one can say anything about reliability of the models.