= Solution
The <profile likelihood> for the parameter of interest replaces the <nuisance parameter> by its constrained best fit: $L_p(\psi)=\sup_\lambda L(\psi,\lambda)$, with $\ell_p(\psi)=\ell(\psi,\widehat\lambda_\psi)$. It is not in general a normalized sampling density. A <conditional likelihood> instead uses the sampling law conditional on a statistic carrying the nuisance dependence. In the canonical <exponential family> here, conditioning on $S$ removes $\lambda$ because the factor involving $\lambda$ is constant over the conditioning fibre.
Let $\bar T_1=n^{-1}\sum_i\tau_1(y_i)$ and let the observed $S$ equal $s$. For the <cumulant-generating function> of $\tau_2(Y)$ at fixed $\psi$,
$$
K(t)=d(\psi,\lambda+t)-d(\psi,\lambda).
$$
The saddlepoint solves $K'(\widehat t)=s$, equivalently $d_\lambda(\psi,\widehat\lambda_\psi)=s$ with $\widehat\lambda_\psi=\lambda+\widehat t$. This is also the constrained <likelihood> equation, since $\ell_\lambda=n(s-d_\lambda)$. Assume an interior saddle, $d_{\lambda\lambda}>0$, and the usual smooth nonlattice conditions under which $S$ has a density. <Exponential tilting> to $\widehat\lambda_\psi$ makes $s$ the tilted mean. A local <normal approximation> gives tilted density $\{n/(2\pi d_{\lambda\lambda}(\psi,\widehat\lambda_\psi))\}^{1/2}$ at that mean; undoing the tilt gives the <saddlepoint density approximation>
$$
\boxed{\widehat f_S(s;\psi,\lambda)=\left[\frac{n}{2\pi d_{\lambda\lambda}(\psi,\widehat\lambda_\psi)}\right]^{1/2}
\exp\left\{n\left[d(\psi,\widehat\lambda_\psi)-d(\psi,\lambda)-(\widehat\lambda_\psi-\lambda)s\right]\right\}.}
$$
The factor $n^{1/2}$ is for the <sample mean>, rather than the sum. A transform-based derivation gives the same expression by expanding $K$ to second order at its stationary point.
The joint <log-likelihood> is $n\psi\bar T_1+n\lambda s-nd(\psi,\lambda)-\sum_iQ(y_i)$. Subtracting $\log\widehat f_S(s;\psi,\lambda)$ cancels every appearance of the original $\lambda$, leaving
$$
\widehat\ell_c(\psi)=n\psi\bar T_1+n\widehat\lambda_\psi s-nd(\psi,\widehat\lambda_\psi)-\sum_iQ(y_i)
+\frac12\log d_{\lambda\lambda}(\psi,\widehat\lambda_\psi)-\frac12\log\frac n{2\pi}.
$$
Any change from the ambient density to a conditional density on $S=s$ contributes only a data-dependent <Jacobian determinant>. Thus, up to a constant <independent> of $\psi$, the <saddlepoint conditional likelihood adjustment> is
$$
\boxed{\widehat\ell_c(\psi)=\ell_p(\psi)+B(\psi),\qquad B(\psi)=\frac12\log d_{\lambda\lambda}(\psi,\widehat\lambda_\psi).}
$$
Equivalently $B=\tfrac12\log j_{\lambda\lambda}(\psi,\widehat\lambda_\psi)$ up to the constant $-\tfrac12\log n$, where $j_{\lambda\lambda}=nd_{\lambda\lambda}$ is the nuisance <observed information>. The adjustment has a positive sign. The printed exponential-family form alone does not guarantee a density for $S$: for example a continuous $Y$ can have $\tau_2(Y)=\mathbf1_{\{Y>0\}}$. In a lattice case one must approximate the probability mass of the sum instead; the displayed density calculation uses the regular continuous interpretation.
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