Mankiw's version of the Cagan model - need help interpreting it
I think that you are unnecessarily overthinking it. For any relationship of the form:
$$\ln y = \beta \ln x  $$
the interpretation of beta coefficient is that $1\%$ increase in $x$ leads to $\beta$ $\%$ increase in $y$. The mathematical reason why this relationship holds was already explored at cross-validated stack exchange and you can see it here, or on this website or in virtually any econometric textbook since the log-log form is important, so I wont be unnecessarily restating it.
Mankiw defines real money balances as $M/P$ or in logs $m-p$, hence the left hand side of your equation is by definition the real money balance term expressed in logs $\ln (M_t/P_t)$. The inflation is by definition change in price level again in your case expressed in logs $\ln(P_{t+1}/P_t)$. So you can directly apply the interpretation from previous paragraph as Mankiw does.
Furthermore, you can actually derive it also with your calculations. To be more specific. Growth rate $g$ for variable $x$ is given as $g_x= \frac{x_{t+1}-x_t}{x_t}$. Then we know that:
$$\ln x_{t+1} = \ln ((1+g_x)x_t) \implies \ln x_{t+1} = \ln (1+g_x)+ \ln x_t  $$
Then since $\ln (1+g_x) \approx g_x$ we have:
$$\ln x_{t+1} = g_x  + \ln x_t \implies \ln x_{t+1} -\ln x_t =g_x $$
as a consequence you can actually directly say that:
$$\ln(M_t/P_t)=-\gamma \ln(P_{t+1}/P_t) \approx \ln(M_t/P_t)=-\gamma \left( \frac{P_{t+1}-P_t}{P_t} \right)$$
that 1 in your final expression should not be there. In this case the LHS gives you percentage in the real money balances and LHS percentage change in price level (inflation).