Step 1:
Padarthadharmasangraha was written by:
\[
\boxed{\text{Prashastapada}}
\]
Thus:
\[
A \rightarrow III
\]
Step 2:
Vedantasara is authored by:
\[
\boxed{\text{Sadananda}}
\]
Thus:
\[
B \rightarrow IV
\]
Step 3:
Tattvavaisharadi is by Vachaspati Mishra, while Tantravarttika is by Kumaril Bhatta.
\[
C \rightarrow I,\quad D \rightarrow II
\]
Step 4:
Hence, the correct sequence is:
\[
A-III,\; B-IV,\; C-I,\; D-II
\]
Therefore, the correct answer is:
\[
\boxed{\text{(2)}}
\]