Asymptotic Distribution of the LS Estimator using an Extra Ratio of the Coefficients

Assume $\boldsymbol{y}=\boldsymbol{\iota}\beta_1+\boldsymbol{x}\beta_2+\boldsymbol{u}$ where $\boldsymbol\iota$ is the n-vector of ones and $\{u_i\}$ are i.i.d. with $E(u_i)=0$ and $E(u_i^2)=\sigma^2$. Now, assume that $\boldsymbol{x’x}/n\to c>0$ and $\boldsymbol{\iota’x}/n\to 0$ as $n\to\infty.$ Suppose there is an estimator $\hat\gamma$ (independent from $\boldsymbol{u}$) for the ratio of the coefficients $\gamma=\beta_1/\beta_2$, and it follows that
$$
\sqrt{n}(\hat\gamma-\gamma)\overset{A}{\sim}\mathcal{N}(0,\lambda^2).
$$
Define
$$
\hat\beta_2=\frac{(\boldsymbol{x}+\boldsymbol{\iota}\hat\gamma)’\boldsymbol{y}}{(\boldsymbol{x}+\boldsymbol{\iota}\hat\gamma)'(\boldsymbol{x}+\boldsymbol{\iota}\hat\gamma)}.
$$
What is the asymptotic distribution of this estimator? I’ve tried to decompose $\hat\beta_2$ but however I do that I end up with correlation problem, i.e. I cannot derive the asymptotic distribution of the sum of all parts from decomposition.

Any suggestion is appreciated.

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