For a sample of $n$ observations on $(X_i,Y_i)$, the likelihood function given $(x_1,y_1),\ldots,(x_n,y_n)$ is
\begin{align}
L(\theta)&=\prod_{i=1}^n e^{-\left(x_i/\theta+\theta y_i\right)}\mathbf1_{x_i>0,y_i>0}
\\&=\exp\left[{-\sum_{i=1}^n\left(\frac{x_i}{\theta}+\theta y_i\right)}\right]\mathbf1_{x_1,\ldots,x_n,y_1,\ldots,y_n>0}
\\&=\exp\left[-\frac{n\bar x}{\theta}-\theta n\bar y\right]\mathbf1_{x_{(1)},y_{(1)}>0}\quad,\,\theta>0
\end{align}
This, as you say, yields the MLE $$\hat\theta(\mathbf X,\mathbf Y)=\sqrt{\frac{\overline X}{\overline Y}}$$, where $\overline X,\overline Y$ are the sample means.
Now $X_i$'s are i.i.d $\mathsf{Exp}$ with mean $\theta$, and $Y_i$'s are i.i.d $\mathsf{Exp}$ with mean $1/\theta$, where $X_i,Y_i$ are independent.
Or, $$\frac{2}{\theta}X_i\stackrel{\text{ i.i.d }}\sim\chi^2_2\quad,\text{ independent of }\quad2\theta Y_i\stackrel{\text{ i.i.d }}\sim\chi^2_2$$
Therefore summing over the observations, we have
$$\frac{2}{\theta}n\overline X\sim\chi^2_{2n}\quad,\text{ independent of }\quad 2\theta n\overline Y\sim\chi^2_{2n}$$
Taking ratio of the statistics gives $$\frac{\overline X}{\theta^2 \overline Y}\sim F_{2n,2n}$$
So the (exact) distribution of the MLE $\hat\theta$ can be expressed as $$\frac{\hat\theta}{\theta}\stackrel{d}{=} \sqrt F\quad,\text{ where }F\sim F_{2n,2n}$$
Since by (weak) law of large numbers $\frac{1}{n}\sum\limits_{i=1}^n X_i\stackrel{P}\longrightarrow E(X_1)=\theta$ and $\frac{1}{n}\sum\limits_{i=1}^n Y_i\stackrel{P}\longrightarrow E(Y_1)=\frac{1}{\theta}$,
$$\hat\theta^2=\frac{\frac{1}{n}\sum\limits_{i=1}^n X_i}{\frac{1}{n}\sum\limits_{i=1}^n Y_i}\stackrel{P}\longrightarrow \frac{E(X_1)}{E(Y_1)}=\theta^2$$
And by continuous mapping theorem, $$\hat\theta \stackrel{P}\longrightarrow \theta$$
This further implies convergence in distribution. So as one might expect, the asymptotic distribution of $\hat\theta$ is degenerate at $\theta$.
Asymptotic normality of MLE is a general result which holds under some smoothness assumptions of the population density. These so called regularity conditions are sufficient conditions for verifying asymptotic normality. If these conditions hold, then indeed a non-degenerate limiting distribution of MLE is given by
$$\sqrt n\left(\hat\theta-\theta\right)\stackrel{L}\longrightarrow N\left(0,\frac1{I(\theta)}\right)\,,$$
where $I(\theta)$ is the Fisher information contained in a single observation.
For a direct proof of asymptotic normality in this particular problem, see this answer.
Alternatively, we can use the multivariate central limit theorem combined with the delta method to reach the same answer.
By CLT we have
$$\sqrt n\left(\left(\overline X,\overline Y\right)-\left(\theta,\frac1{\theta}\right)\right)\stackrel{L}\longrightarrow N\left(\mathbf 0,\Sigma\right)\,,$$
where $$\Sigma=\begin{pmatrix}\theta^2 & 0 \\ 0 & \frac1{\theta^2}\end{pmatrix}$$
By the delta-method, the following holds for some function $g$ :
$$\sqrt n\left(g\left(\overline X,\overline Y\right)-g\left(\theta,\frac1{\theta}\right)\right)\stackrel{L}\longrightarrow N\left( 0, \nabla g\left(\theta,\frac1{\theta}\right)^T \cdot \Sigma \cdot \nabla g\left(\theta,\frac1{\theta}\right)\right)$$
Taking $g(x,y)=\sqrt{\frac xy}$ for $x,y>0$, this reduces to
$$\sqrt n\left(\hat\theta-\theta\right)\stackrel{L}\longrightarrow N\left(0,\frac{\theta^2}{2}\right)$$
