Well, let's see.
The Lagrangian is $$ L~=~ \frac{\langle \psi | i \partial_t -H |\psi\rangle}{2||\psi ||^2} ~+~ c.c.,\qquad ||\psi ||^2~\equiv~\langle \psi | \psi\rangle. \tag{A}$$ An infinitesimal variation is $$\begin{align} \delta L~=~&\frac{\langle \delta \psi | i \partial_t -H |\psi\rangle + \langle \psi | i \partial_t -H |\delta \psi\rangle}{2||\psi ||^2} \cr &+ \langle \psi | i \partial_t -H |\psi\rangle\delta \frac{1}{2||\psi ||^2} ~+~ c.c.\cr ~\stackrel{IBP}{\sim}~&\frac{\langle \delta \psi | i \partial_t -H |\psi\rangle + \langle \psi |- i \stackrel{\leftarrow}{\partial_t} -H |\delta \psi\rangle}{2||\psi ||^2} \cr &-\langle \psi | \delta \psi\rangle \partial_t \frac{i}{2||\psi ||^2} \cr & -\langle \psi | i \partial_t -H |\psi\rangle \frac{\langle \delta\psi | \psi\rangle+\langle \psi | \delta \psi\rangle }{2||\psi ||^4} ~+~ c.c.\cr ~=~&\frac{\langle \delta \psi | i \partial_t -H |\psi\rangle}{||\psi ||^2} + \langle \delta\psi | \psi\rangle \partial_t \frac{i}{2||\psi ||^2} \cr & -\langle\delta\psi |\psi\rangle \frac{\langle \psi | i \partial_t -H |\psi\rangle + \overline{\langle \psi | i \partial_t -H |\psi\rangle}}{2||\psi ||^4} ~+~ c.c. \end{align}\tag{B}$$ In the 2nd equality of eq. (B) we dropped a total time derivative term. In the 3rd equality of eq. (B) we strategically swopped terms with the $c.c.$ part.
We can treat $\langle \delta \psi |$ and $| \delta \psi |\rangle$ as independent in eq. (B), cf. e.g. this Phys.SE post. This leads to$^1$ $$\begin{align} 0~\approx~& ( i \partial_t -H )|\psi\rangle + | \psi\rangle ||\psi ||^2 \partial_t \frac{i}{2||\psi ||^2} \cr & -| \psi\rangle \frac{\langle \psi | i \partial_t -H |\psi\rangle + \overline{\langle \psi | i \partial_t -H |\psi\rangle}}{2||\psi ||^2}. \end{align}\tag{C}$$ Contracting with $\langle \psi |$ yields $$ 0~\approx~ \partial_t \frac{i}{2||\psi ||^2} + \frac{\langle \psi | i \partial_t -H |\psi\rangle - \overline{\langle \psi | i \partial_t -H |\psi\rangle}}{2||\psi ||^4}. \tag{D}$$ Eqs. (C) & (D) indeed imply the projected TDSE in the book $$ 0~\approx~ ( i \partial_t -H )|\psi\rangle -|\psi\rangle \frac{\langle \psi | i \partial_t -H |\psi\rangle}{||\psi ||^2}.\tag{2.3} $$
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$^1$ The $\approx$ symbol means equality modulo EL equations.