Here's an elementary proof that $2n-2$ is a lower bound.

Suppose that $$V_1\xrightarrow{\alpha_1}V_2\xrightarrow{\alpha_2}\dots\xrightarrow{\alpha_{n-2}}V_{n-1}\xrightarrow{\alpha_{n-1}}V_n$$ is a representation of the linearly ordered $A_n$ quiver $Q$ that is a representation of $FQ/I$, where $I$ is the one-dimensional ideal of the path algebra $FQ$ generated by the longest path, so that $\alpha_{n-1}\dots\alpha_1=0$, and that it is faithful as a representation of $FQ/I$.

By faithfulness, $\alpha_{n-2}\dots\alpha_1(v_1)\neq0$ for some $v_1\in V_1$. For $1<i<n$ let $v_i=\alpha_i\dots\alpha_1(v_1)\in V_i$.

Also by faithfulness, $\alpha_{n-1}\dots\alpha_2(w_2)\neq0$ for some $w_2\in V_2$. For $2<i\leq n$ let $w_i=\alpha_i\dots\alpha_2(w_2)$.

For $1<i<n$, $v_i$ and $w_i$ are linearly independent (consider their images in $V_{n-1}$ and $V_n$), so $v_1,\dots,v_{n-1},w_2,\dots,w_n$ span a ($2n-2$)-dimensional subrepresentation (which is isomorphic to the representation described in the OP).

Here's an elementary proof that $2n-2$ is a lower bound.

Suppose that $$V_1\xrightarrow{\alpha_1}V_2\xrightarrow{\alpha_2}\dots\xrightarrow{\alpha_{n-2}}V_{n-1}\xrightarrow{\alpha_{n-1}}V_n$$ is a representation of the linearly ordered $A_n$ quiver $Q$ that is a representation of $FQ/I$, where $I$ is the one-dimensional ideal of the path algebra $FQ$ generated by the longest path, so that $\alpha_{n-1}\dots\alpha_1=0$, and that it is faithful as a representation of $FQ/I$.

By faithfulness, $\alpha_{n-2}\dots\alpha_1(v_1)\neq0$ for some $v_1\in V_1$. For $1<i<n$ let $v_i=\alpha_{i-1}\dots\alpha_1(v_1)\in V_i$.

Also by faithfulness, $\alpha_{n-1}\dots\alpha_2(w_2)\neq0$ for some $w_2\in V_2$. For $2<i\leq n$ let $w_i=\alpha_{i-1}\dots\alpha_2(w_2)\in V_i$.

For $1<i<n$, $v_i$ and $w_i$ are linearly independent (consider their images in $V_{n-1}$ and $V_n$), so $v_1,\dots,v_{n-1},w_2,\dots,w_n$ span a ($2n-2$)-dimensional subrepresentation (which is isomorphic to the representation described in the OP).

Here's an elementary proof that $2n-2$ is a lower bound.

Suppose $$V_1\xrightarrow{\alpha_1}V_2\xrightarrow{\alpha_2}\dots\xrightarrow{\alpha_{n-2}}V_{n-1}\xrightarrow{\alpha_{n-1}}V_n$$ is a representation of the linearly ordered $A_n$ quiver $Q$ that is a representation of $FQ/I$, where $I$ is the one-dimensional ideal of the path algebra $FQ$ generated by the longest path, so that $\alpha_{n-1}\dots\alpha_1=0$, and that it is faithful as a representation of $FQ/I$.

By faithfulness, $\alpha_{n-2}\dots\alpha_1(v_1)\neq0$ for some $v_1\in V_1$. For $1<i<n$ let $v_i=\alpha_i\dots\alpha_1(v_1)\in V_i$.

Also by faithfulness, $\alpha_{n-1}\dots\alpha_2(w_2)\neq0$ for some $w_2\in V_2$. For $2<i\leq n$ let $w_i=\alpha_i\dots\alpha_2(w_2)$.

For 1<i<n, $v_i$ and $w_i$ are linearly independent (consider their images in $V_{n-1}$ and $V_n$), so $v_1,\dots,v_{n-1},w_2,\dots,w_n$ span a $2n-2$-dimensional subrepresentation (which is isomorphic to the representation described in the OP).

Here's an elementary proof that $2n-2$ is a lower bound.

Suppose that $$V_1\xrightarrow{\alpha_1}V_2\xrightarrow{\alpha_2}\dots\xrightarrow{\alpha_{n-2}}V_{n-1}\xrightarrow{\alpha_{n-1}}V_n$$ is a representation of the linearly ordered $A_n$ quiver $Q$ that is a representation of $FQ/I$, where $I$ is the one-dimensional ideal of the path algebra $FQ$ generated by the longest path, so that $\alpha_{n-1}\dots\alpha_1=0$, and that it is faithful as a representation of $FQ/I$.

By faithfulness, $\alpha_{n-2}\dots\alpha_1(v_1)\neq0$ for some $v_1\in V_1$. For $1<i<n$ let $v_i=\alpha_i\dots\alpha_1(v_1)\in V_i$.

Also by faithfulness, $\alpha_{n-1}\dots\alpha_2(w_2)\neq0$ for some $w_2\in V_2$. For $2<i\leq n$ let $w_i=\alpha_i\dots\alpha_2(w_2)$.

For $1<i<n$, $v_i$ and $w_i$ are linearly independent (consider their images in $V_{n-1}$ and $V_n$), so $v_1,\dots,v_{n-1},w_2,\dots,w_n$ span a ($2n-2$)-dimensional subrepresentation (which is isomorphic to the representation described in the OP).