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• 545
• 1
• 5
• 14

Jelly, 37 bytes

2*’
S€=9a×ばつ2*9¤ʋ
Ñe3ƊƇμ’3œcç"Ñ’$F1Ƈ

Try it online!

Also a mess.

Link 1: computes \2ドル^x-1\$

2 Two
 * To the power of the input
 ’ Decrement

Link 2: given a list of subsets of length \$p-1\$ of the input list, and \$q-1\$ (as defined in the question), return a list with all valid answers or 0.

S€=9a×ばつ2*9¤ʋ 
 € For each subset:
S Compute the sum
 = Check if equal to... (vectorizes)
 9 ...q-1
 a Logical AND with...
 ʋ ...the four previous links as a dyad
 Ç The previous link as a monad (2^x-1 for each x in each subset)
 € For each subset:
 P Compute the product
 ×ばつ Multiply by...
 ¤ ...a nilad followed by some links, as a nilad:
 2 Two
 * To the power of:
 9 q-1

Link 3 (main link): given the input list, outputs all valid answers.

Ñe3ƊƇμ’3œcç"Ñ’$F1Ƈ
 Ƈ Filter the list using:
 Ɗ The last three links as a monad
Ñ The next link as a monad (wraps around to link 1)
 e Check if it's present in...
 3 ...the input list
 μ Start a new monadic chain
(At this point, the list contains all 'p' in the input list such that 2^p-1 is also in the input list.)
 ’ Decrement
 œc All combinations of ... of any length in the list (vectorizes over the right argument)
 3 The input list
 " Zipping over the left and right arguments:
 ç Call the previous link as a dyad, with the next link as the right argument
 $ The last two links as a monad:
 Ñ The next link (wraps around to link 1) on each valid 'p', thus generating all the candidates for 'q'
 ’ Decrement (thus generating all the candidates for q-1)
 F Flatten
 Ƈ Filter the list using:
 1 The truth value of each element itself (thus removing all falsy values from the list)