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You wish to approximate $$p_{\rm exact}(\rho)=e^{-\rho^2-\rho_0^2} \rho I_0(2\rho \rho_0).$$

For small $\rho_0$ a Taylor series in powers of $\rho_0$ is accurate, $$p_{\rm small}(\rho)=e^{-\rho^2} \rho (1+\rho^2 \rho_0^2-\rho_0^2).$$ For large $\rho_0$ an asymptotic expansion of the Bessel function gives $$p_{\rm large}(\rho)=\frac{1}{2\sqrt{\pi}}e^{-\rho^2-\rho_0^2}e^{2\rho\rho_0}.$$

In the plots below I compare $\int_0^x p(\rho)\,d\rho$ for the exact expression (blue) and the approximation (orange). It turns out the approximations $p_{\rm small}$ and $p_{\rm large}$ are already quite accurate for $\rho_0\lesssim 0.5$ and $\rho_0\gtrsim 3$, respectively.

Q: The OP seeks a "reasonable approximation" for large or small $\rho_0$ of the function $$P(x)= \int_0^{x} e^{-\rho^2-\rho_0^2} \rho I_0(2\rho \rho_0)\,d\rho,\;\;x\geq 0.$$

Consider the kernel $$p_{\rm exact}(\rho)=e^{-\rho^2-\rho_0^2} \rho I_0(2\rho \rho_0).$$ For small $\rho_0$ a Taylor series in powers of $\rho_0$ is accurate, $$p_{\rm small}(\rho)=e^{-\rho^2} \rho (1+\rho^2 \rho_0^2-\rho_0^2).$$ For large $\rho_0$ an asymptotic expansion of the Bessel function gives $$p_{\rm large}(\rho)=\frac{1}{2\sqrt{\pi}}e^{-\rho^2-\rho_0^2}e^{2\rho\rho_0}.$$

In the plots below I compare $\int_0^x p(\rho)\,d\rho$ for the exact expression (blue) and the approximation (orange). It turns out the approximations $p_{\rm small}$ and $p_{\rm large}$ are already quite accurate for $\rho_0\lesssim 0.5$ and $\rho_0\gtrsim 3$, respectively.

You wish to approximate $$p_{\rm exact}(\rho)=e^{-\rho^2-\rho_0^2} \rho I_0(2\rho \rho_0).$$

For small $\rho_0$ a Taylor series in powers of $\rho_0$ is accurate, $$p_{\rm small}=e^{-\rho^2} \rho (1+\rho^2 \rho_0^2-\rho_0^2).$$ For large $\rho_0$ an asymptotic expansion of the Bessel function gives $$p_{\rm large}(\rho_0)=\frac{1}{2\sqrt{\pi}}e^{-\rho^2-\rho_0^2}e^{2\rho\rho_0}.$$

In the plots below I compare $\int_0^x p(\rho)\,d\rho$ for the exact expression (blue) and the approximation (orange). It turns out the approximations $p_{\rm small}$ and $p_{\rm large}$ are already quite accurate for $\rho_0\lesssim 0.5$ and $\rho_0\gtrsim 3$, respectively.

You wish to approximate $$p_{\rm exact}(\rho)=e^{-\rho^2-\rho_0^2} \rho I_0(2\rho \rho_0).$$

For small $\rho_0$ a Taylor series in powers of $\rho_0$ is accurate, $$p_{\rm small}(\rho)=e^{-\rho^2} \rho (1+\rho^2 \rho_0^2-\rho_0^2).$$ For large $\rho_0$ an asymptotic expansion of the Bessel function gives $$p_{\rm large}(\rho)=\frac{1}{2\sqrt{\pi}}e^{-\rho^2-\rho_0^2}e^{2\rho\rho_0}.$$

In the plots below I compare $\int_0^x p(\rho)\,d\rho$ for the exact expression (blue) and the approximation (orange). It turns out the approximations $p_{\rm small}$ and $p_{\rm large}$ are already quite accurate for $\rho_0\lesssim 0.5$ and $\rho_0\gtrsim 3$, respectively.

You wish to approximate $$p_{\rm exact}(\rho)=e^{-\rho^2-\rho_0^2} \rho I_0(2\rho \rho_0).$$ For $\rho_0\ll 1$ a Taylor series in powers of $\rho_0$ is accurate. Let me consider the opposite regime of large $\rho_0$. An asymptotic expansion of the Bessel function gives $$p_{\rm approximate}(\rho_0)=\frac{1}{2\sqrt{\pi}}e^{-\rho^2-\rho_0^2}e^{2\rho\rho_0}.$$ This large-$\rho_0$ approximation to $\int_0^x p(\rho)\,d\rho$ is already quite accurate for $\rho_0= 3$, see the plot (blue = exact, orange = approximate, almost indistuishable for $\rho_0=3$).

You wish to approximate $$p_{\rm exact}(\rho)=e^{-\rho^2-\rho_0^2} \rho I_0(2\rho \rho_0).$$

For small $\rho_0$ a Taylor series in powers of $\rho_0$ is accurate, $$p_{\rm small}=e^{-\rho^2} \rho (1+\rho^2 \rho_0^2-\rho_0^2).$$ For large $\rho_0$ an asymptotic expansion of the Bessel function gives $$p_{\rm large}(\rho_0)=\frac{1}{2\sqrt{\pi}}e^{-\rho^2-\rho_0^2}e^{2\rho\rho_0}.$$

In the plots below I compare $\int_0^x p(\rho)\,d\rho$ for the exact expression (blue) and the approximation (orange). It turns out the approximations $p_{\rm small}$ and $p_{\rm large}$ are already quite accurate for $\rho_0\lesssim 0.5$ and $\rho_0\gtrsim 3$, respectively.