1. Use determinant invariance under similarity

We need to find $\left|P^{-1}AP-2I\right|.$

Observe that $P^{-1}AP-2I=P^{-1}AP-2P^{-1}IP=P^{-1}(A-2I)P.$

Hence, $\det(P^{-1}AP-2I)=\det\big(P^{-1}(A-2I)P\big).$

Using multiplicative property of determinant, $\det(P^{-1}(A-2I)P)=\det(P^{-1})\det(A-2I)\det(P)=\det(A-2I).$

So it is enough to compute $|A-2I|.$

2. Compute $A-2I$

Given $A=\begin{bmatrix}2&1&2\\6&2&11\\3&3&2\end{bmatrix},$ we get $A-2I=\begin{bmatrix}0&1&2\\6&0&11\\3&3&0\end{bmatrix}.$

3. Find its determinant

Expand along the first row: $\det(A-2I)=0\cdot\begin{vmatrix}0&11\\3&0\end{vmatrix}-1\cdot\begin{vmatrix}6&11\\3&0\end{vmatrix}+2\cdot\begin{vmatrix}6&0\\3&3\end{vmatrix}.$

Now, $\begin{vmatrix}6&11\\3&0\end{vmatrix}=6\cdot 0-11\cdot 3=-33,$ so $-1\cdot(-33)=33.$

Also, $\begin{vmatrix}6&0\\3&3\end{vmatrix}=6\cdot 3-0\cdot 3=18,$ so $2\cdot 18=36.$

Therefore, $\det(A-2I)=33+36=69.$

Thus, $\left|P^{-1}AP-2I\right|=69.$

4. Prime factors of 69

$69=3\times 23.$

Sum of prime factors: $3+23=26.$

5. Final answer

The required sum is $\boxed{26}.$ So the correct option is D.

6. Comparison with stored answer

Stored correct answer: D

Our derived answer: D (26)

Hence, the derived answer agrees with the stored answer.