The equation you found, \begin{equation} \int_{r=0}^\infty \int_{\theta=0}^{\pi/n} e^{i n \theta} p_n(r)\,r\,dr\,d\theta = \int_{r=0}^\infty \int_{\theta=0}^{\pi/m} e^{i m \theta} p_m(r)\,r\,dr\,d\theta,\tag{1} \end{equation} (which happens to be true in a restricted sense, see below) does not imply $\int_{0}^\infty p_n(r)\,r\,dr = \int_{0}^\infty p_m(r)\,r\,dr$. Just do the angular integrals. You will find instead that it implies $\frac{1}{n} \int_{0}^\infty p_n(r)\,r\,dr = \frac{1}{m}\int_{0}^\infty p_m(r)\,r\,dr.$ We can check this explicitly, $\begin{eqnarray} \frac{1}{n} \int_{0}^\infty p_n(r)\,r\,dr &=& \frac{1}{n} \int_0^\infty P(k)\,k \,dk \int_0^\infty J_n(k r)\,r\,dr \\ &=& \frac{1}{n} \int_0^\infty P(k)\,k \,dk \frac{n}{k^2} \\ &=& \int_0^\infty P(k)\,k^{-1} \,dk. \end{eqnarray}$ The final integral is just some constant, independent of $n$. $P(k)$ must of course be well enough behaved so the integral makes sense.
Here we have used the fact that the Hankel transform of $r^s$ is $\frac{\Gamma\left(1+\frac{1}{2}(n+s)\right)}{\Gamma\left(\frac{1}{2}(n-s)\right)} \frac{2^{s+1}}{k^{s+2}}.$ Thus, for example, the Hankel transform of $1$ (used above) is $n/k^2$.
Addendum: Notice that, strictly speaking, the integral $\int_0^\infty r^{s+1} J_n(k x) dr = \frac{\Gamma\left(1+\frac{1}{2}(n+s)\right)}{\Gamma\left(\frac{1}{2}(n-s)\right)} \frac{2^{s+1}}{k^{s+2}}$ is divergent unless $-\mathrm{Re}\, n-2< \mathrm{s} <-\frac{1}{2}$ and $k>0$. These conditions are not satisfied for $s=0$. However, we can define the value of the integral to be the analytic continuation of the above formula to $s=0$. In fact, this practice is quite common and useful in physics. Thus, strictly speaking each side of (1) is divergent, but in a sense it is still true!