We'll show a more general statement. Suppose $(X,\Delta)$ has klt singularities and $f : Y \to X$ is a projective birational morphism with $Y$ normal and $\mathbb{Q}$-factorial. Suppose further that $f$ is not small so that $Ex(f)$ contains some divisor.

Claim: Then $K_Y + f_*^{-1}\Delta + E$ is not $f$-nef where $E$ is the reduced exceptional divisor.

Indeed by the definition of klt, we have

$$ K_Y + f_*^{-1}\Delta + E \equiv f^*(K_X + \Delta) + \sum a_i E_i $$ where $a_i > 0$ and the sum runs over prime $f$-exceptional divisors. Let us denote $-B \colon = \sum a_i E_i$. Then for any $f$-exceptional curve $C$, we have $$ (K_Y + f_*^{-1}\Delta + E).C = (-B).C. $$

Now we apply the following negativity lemma (see for example Lemma 3.39 Kollár-Mori).

Negativity Lemma: Let $f : Y \to X$ be a proper birational morphism between normal varieties. Suppose $-B$ is an $f$-nef $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor. Then $B$ is effective if and only if $f_*B$ is effective.

In our case, $B$ is $f$-exceptional so $f_*B = 0$ is effective. Thus if $-B$ were nef, this would contradict that $a_i > 0$. This proves the claim.

Now if $X$ is $\mathbb{Q}$-factorial, then the exceptional locus of any projective birational $f : Y \to X$ contains a divisor. In fact the exceptional locus is pure codimension $1$ (see for example Corollary 2.63 in Kollár-Mori). Therefore in the setting of the question, if the MMP relative $X$ terminates, it has to terminate with $X$ itself.

Note that without the $\mathbb{Q}$-factoriality assumption on $X$, the statement is false in general because the exceptional locus of the resolution may not be pure codimension $1$. In this case, the MMP terminates in a log minimal model $(Z, \Delta_Z)$ with a small contraction $q : Z \to X$ where $q^*(K_X + \Delta) = K_Z + \Delta_Z$ if $q$-nef.

We'll show a more general statement. Suppose $(X,\Delta)$ has klt singularities and $f : Y \to X$ is a projective birational morphism with $Y$ normal and $\mathbb{Q}$-factorial. Suppose further that $f$ is not small so that $Ex(f)$ contains some divisor.

Claim: Then $K_Y + f_*^{-1}\Delta + E$ is not $f$-nef where $E$ is the reduced exceptional divisor.

Indeed by the definition of klt, we have

$$ K_Y + f_*^{-1}\Delta + E \equiv f^*(K_X + \Delta) + \sum a_i E_i $$ where $a_i > 0$ and the sum runs over prime $f$-exceptional divisors. Let us denote $-B \colon = \sum a_i E_i$. Then for any $f$-exceptional curve $C$, we have $$ (K_Y + f_*^{-1}\Delta + E).C = (-B).C. $$

Now we apply the following negativity lemma (see for example Lemma 3.39 in Kollár-Mori).

Negativity Lemma: Let $f : Y \to X$ be a proper birational morphism between normal varieties. Suppose $-B$ is an $f$-nef $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor. Then $B$ is effective if and only if $f_*B$ is effective.

In our case, $B$ is $f$-exceptional so $f_*B = 0$ is effective. Thus if $-B$ were nef, this would contradict that $a_i > 0$. This proves the claim.

Now if $X$ is $\mathbb{Q}$-factorial, then the exceptional locus of any projective birational $f : Y \to X$ contains a divisor. In fact the exceptional locus is pure codimension $1$ (see for example Corollary 2.63 in Kollár-Mori). Therefore in the setting of the question, if the MMP relative $X$ terminates, it has to terminate with $X$ itself.

Note that without the $\mathbb{Q}$-factoriality assumption on $X$, the statement is false in general because the exceptional locus of the resolution may not be pure codimension $1$. In this case, the MMP terminates in a log minimal model $(Z, \Delta_Z)$ with a small contraction $q : Z \to X$ where $q^*(K_X + \Delta) = K_Z + \Delta_Z$ is $q$-nef.