I wouldn't expect any of these bounds to give really superb estimates of the last time your ratio achieves a certain record because $E(r)$ and your ratio undergo rapid fluctuations. Also, the exciting places are unlikely to happen at integers.
The last integer for which $\left| \frac{\pi r^2 }{\pi r^2 + E(r)} - 1 \right| \gt 10^{-2}$ is $r=12.$ However $N(\sqrt{586})=1861$ while $586\pi\approx 1840.97$ giving a ratio of $0.98923=1-0.01076.$ If you change $0.01$ to $0.0132$ the record is at $r=\sqrt{425}.$ But if you change it to $0.0082$ it is at $r=\sqrt{986}.$
Are you considering all positive real $r$ or all$ \sqrt{a^2+b^2}$ or only integers? The local minima of $E(r)$ occur at the places where $r=\sqrt{a^2+b^2}.$ Then $E(r)$ increases (at a rate of $2\pi r$) until the next local minimum.
For $r=\sqrt{281},\sqrt{288},\sqrt{289}$ one has $N(r)=885,889,901$ and the last time the ratio is as large as $1.01$ is at $\sqrt{288}$ (or $\sqrt{288}-\varepsilon$ if allowed) with ratio of $1.0177$ (or $1.0223$ .)