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The answer is yes, assuming the existence of $\aleph_2$-many measurable cardinals. To see this, assume $GCH+\Diamond$ holds and $S$ is a discrete set of measurable cardinals of size $\aleph_2.$

Step 1. Force with Prikry product forcing $P_S$ to change the cofinality of each element of $S$ into $\omega.$

Note that the extension is of the form $V[(x_\alpha: \alpha\in S)]$, where each $x_\alpha$ is an $\omega-$sequence cofinal in $\alpha.$

Step 2. Force with Jensen's coding theorem, to code everything into a real $r$, so that we have $V[(x_\alpha: \alpha\in S)][r]=V[r]$ (and in fact $=L[r]$).

Step 3. Force over $V[r]$, by a cardinal and $GCH$ preserving forcing iteration to force $\neg \Diamond$.

Note that the generic can be seen as a subset $X$ of $S.$ Now working in $V[r][X],$ define a new sequence $(y_\alpha: \alpha\in S),$ so that $y_\alpha=x_\alpha,$ if $\alpha\in X$ and $y_\alpha=x_\alpha\setminus\{ min(x_\alpha) \}$ if $\alpha\notin X.$

Then let $V_1=V[(y_\alpha: \alpha\in S)]$ and $V_2=V_1[r].$ Note that:

1. $V_1\models GCH+\Diamond,$

2. $V_2=V[(y_\alpha: \alpha\in S)][r]=V[r][X]\models GCH+\neg\Diamond,$

3. $V_2=V_1[r]$, for some real $r$.

The answer is yes, assuming the existence of $\aleph_2$-many measurable cardinals. To see this, assume $GCH+\Diamond$ holds and $S$ is a discrete set of measurable cardinals of size $\aleph_2.$

Step 1. Force with Prikry product forcing $P_S$ to change the cofinality of each element of $S$ into $\omega.$

Note that the extension is of the form $V[(x_\alpha: \alpha\in S)]$, where each $x_\alpha$ is an $\omega-$sequence cofinal in $\alpha.$

Step 2. Force with Jensen's coding theorem, to code everything into a real $r$, so that we have $V[(x_\alpha: \alpha\in S)][r]=V[r]$ (we can do this using a set forcing construction, and assuming that the ground model is a core model).

Step 3. Force over $V[r]$, by a cardinal and $GCH$ preserving forcing iteration to force $\neg \Diamond$.

Note that the generic can be seen as a subset $X$ of $S.$ Now working in $V[r][X],$ define a new sequence $(y_\alpha: \alpha\in S),$ so that $y_\alpha=x_\alpha,$ if $\alpha\in X$ and $y_\alpha=x_\alpha\setminus\{ min(x_\alpha) \}$ if $\alpha\notin X.$

Then let $V_1=V[(y_\alpha: \alpha\in S)]$ and $V_2=V_1[r].$ Note that:

1. $V_1\models GCH+\Diamond,$

2. $V_2=V[(y_\alpha: \alpha\in S)][r]=V[r][X]\models GCH+\neg\Diamond,$

3. $V_2=V_1[r]$, for some real $r$.

I may mention that the above method can be used to prove the consistency of many statements using adding a single real.

The answer is yes, assuming the existence of $\aleph_2$-many measurable cardinals. To see this, assume $GCH+\Diamond$ holds and $S$ is a discrete set of measurable cardinals of size $\aleph_2.$

Step 1. Force with Prikry product forcing $P_S$ to change the cofinality of each element of $S$ into $\omega.$

Note that the extension is of the form $V[(x_\alpha: \alpha\in S)]$, where each $x_\alpha$ is an $\omega-$sequence cofinal in $\alpha.$

Step 2. Force with Jensen's coding theorem, to code everything into a real $r$, so that we have $V[(x_\alpha: \alpha\in S)][r]=V[r]$ (and in fact $=L[r]$).

Step 3. Force over $V[r]$, by a cardinal and $GCH$ preserving forcing iteration to force $\neg \Diamond$.

Note that the generic can be seen as a subset $X$ of $\omega_2.$ Now working in $V[r][X],$ define a new sequence $(y_\alpha: \alpha<\omega),$ so that $y_\alpha=x_\alpha,$ if $\alpha\in X$ and $y_\alpha=x_\alpha\setminus\{ min(x_\alpha) \}$ if $\alpha\notin X.$

Then let $V_1=V[(y_\alpha: \alpha<\omega_1)]$ and $V_2=V_1[r].$ Note that:

1. $V_1\models GCH+\Diamond,$

2. $V_2=V[(y_\alpha: \alpha<\omega_1)][r]=V[r][X]\models GCH+\neg\Diamond,$

3. $V_2=V_1[r]$, for some real $r$.

The answer is yes, assuming the existence of $\aleph_2$-many measurable cardinals. To see this, assume $GCH+\Diamond$ holds and $S$ is a discrete set of measurable cardinals of size $\aleph_2.$

Step 1. Force with Prikry product forcing $P_S$ to change the cofinality of each element of $S$ into $\omega.$

Note that the extension is of the form $V[(x_\alpha: \alpha\in S)]$, where each $x_\alpha$ is an $\omega-$sequence cofinal in $\alpha.$

Step 2. Force with Jensen's coding theorem, to code everything into a real $r$, so that we have $V[(x_\alpha: \alpha\in S)][r]=V[r]$ (and in fact $=L[r]$).

Step 3. Force over $V[r]$, by a cardinal and $GCH$ preserving forcing iteration to force $\neg \Diamond$.

Note that the generic can be seen as a subset $X$ of $S.$ Now working in $V[r][X],$ define a new sequence $(y_\alpha: \alpha\in S),$ so that $y_\alpha=x_\alpha,$ if $\alpha\in X$ and $y_\alpha=x_\alpha\setminus\{ min(x_\alpha) \}$ if $\alpha\notin X.$

Then let $V_1=V[(y_\alpha: \alpha\in S)]$ and $V_2=V_1[r].$ Note that:

1. $V_1\models GCH+\Diamond,$

2. $V_2=V[(y_\alpha: \alpha\in S)][r]=V[r][X]\models GCH+\neg\Diamond,$

3. $V_2=V_1[r]$, for some real $r$.

The following does not answer your question, and in fact it is in the negative direction, but it is too long to be added as a comment.

(A): The answer to your question is no, if $V=L,$ as then for any real $r, L[r]\models \Diamond+CH.$

(B): For your stronger question, again we have the following negative consistency result. Assume that $V$ is a model of $ZFC$ which satisfies the following (such a model exists):

1. Every tree of height and size $\omega_1$ is special,

2. $2^{\aleph_0}=\aleph_2.$

The by a theorem of Todorcevic (see "some combinatorial properties of trees"), any forcing notion which adds a new subset of $\omega_1,$ collapses $\aleph_1$ or $\aleph_2.$ It means that in all generic extensions $V[r]$ of $V$, where $r$ is a new real, the power function changes, and cardinals are collapsed.

These results show that, we need some preparation model to construct your required model $V$.

The answer is yes, assuming the existence of $\aleph_2$-many measurable cardinals. To see this, assume $GCH+\Diamond$ holds and $S$ is a discrete set of measurable cardinals of size $\aleph_2.$

Step 1. Force with Prikry product forcing $P_S$ to change the cofinality of each element of $S$ into $\omega.$

Note that the extension is of the form $V[(x_\alpha: \alpha\in S)]$, where each $x_\alpha$ is an $\omega-$sequence cofinal in $\alpha.$

Step 2. Force with Jensen's coding theorem, to code everything into a real $r$, so that we have $V[(x_\alpha: \alpha\in S)][r]=V[r]$ (and in fact $=L[r]$).

Step 3. Force over $V[r]$, by a cardinal and $GCH$ preserving forcing iteration to force $\neg \Diamond$.

Note that the generic can be seen as a subset $X$ of $\omega_2.$ Now working in $V[r][X],$ define a new sequence $(y_\alpha: \alpha<\omega),$ so that $y_\alpha=x_\alpha,$ if $\alpha\in X$ and $y_\alpha=x_\alpha\setminus\{ min(x_\alpha) \}$ if $\alpha\notin X.$

Then let $V_1=V[(y_\alpha: \alpha<\omega_1)]$ and $V_2=V_1[r].$ Note that:

1. $V_1\models GCH+\Diamond,$

2. $V_2=V[(y_\alpha: \alpha<\omega_1)][r]=V[r][X]\models GCH+\neg\Diamond,$

3. $V_2=V_1[r]$, for some real $r$.