Strömbergsson's footnote refers to $\sum_{|n|\leq N}|c_n|^2$, not $\sum_{|n|\leq N}|n||c_n|^2$. In fact his $c_n$ is Iwaniec's $|n|^{1/2}\hat f_\mathfrak{a}(n)$. Normalization is a serious difficulty in the subject, one must be careful.
Strömbergsson points out that Iwaniec's proof is incorrect, but the result is fine. To fix the given line in Iwaniec's proof, integrate from $|s|/4$ instead of $|s|$, then the claimed bound is all right. This you can see from the discussion below (4.13) in Strömbergsson's paper: already the integral from $|s|/4$ to $|s|/2$ is large. Note that Strömbergsson talks about $s=1/2+iR$ (i.e. the case $\Re s=1/2$), but for the remaining $s$ (i.e. $0\leq s\leq 1$) Iwaniec's bound follows by compactness.
I don't know (from the top of my head) of any better uniform estimate than the one displayed. In special ranges or for Hecke eigenforms one has better estimates, but there are many open questions (e.g. how to give a good lower bound when $N$ is small compared to the various parameters of the form).