= Solution
Introduce an additive background rate $B$, from ambient radiation, detector dark counts or another weakly attenuated contribution. If $S_0$ denotes the unattenuated signal alone, the <attenuation count model with an additive background> is
$$
\boxed{N(n)=S_0e^{-bn}+B.}
$$
It approaches a nonzero constant as the transmitted signal becomes small. The source-off background should ideally be measured in a separate long exposure and its uncertainty included in the fit; otherwise estimate $B\geq0$ jointly using a Poisson count <likelihood>.
There is a normalization detail: the given $N_0$ was itself obtained from an $n=0$ count measurement. If that measurement included the same background, then it measures $S_0+B$, and the <attenuation fit normalized to a total reference rate> is
$$
\boxed{N(n)=(N_0-B)e^{-bn}+B.}
$$
Writing $N_0e^{-bn}+B$ instead treats $N_0$ as a background-subtracted signal calibration. The difference is small here because the background is a few counts per second while $N_0$ is about two thousand, but the distinction prevents double-counting background. The last three observations alone do not give an exact value of $B$, because residual transmitted signal and sizeable counting fluctuations remain.
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