At present, going into the castle would be tantamount to suicide, but on the other hand, you do not wish to be pinned between it and the coming relief force, but you do not wish to lift the siege either.

Thus, there is a simple solution to your problem. You shall build a circumvallation, nothing too fancy, a wooden palisade with some towers. This will at the very least hinder the garrison from sallying forth. To make certain it's extra difficult, you shall have the palisade be preceded by ditches, spikes, and other defensive earthworks, thereby locking them inside of their castle for when the actual battle would come.

If you were to assault the walls right now, it would be 30 DC.

The Svengalian relief force will come in 1/15 turns.

The garrison will run out of supplies in 20 turns.

It is currently turn 1.

The trebuchet has been built. (Will allow you to hurl rocks at the walls.)
It will take 2 turns for additional ladders to be built. (+5 to the assault DC.)
It will take 4 turns for the battering ram to be built. (+10 to the assault DC)
It will take 6 turns for additional artillery to be constructed. (+10 DC when bombarding the walls.)
It will take 8 turns for siege towers to be readied. (+10 to the assault DC)

Roll a1d100, [DC 45] to begin battering the walls.
What shall you do now?

>Use the ballista to start a fire.
>Ask Sophia for magical help.
>Have men start digging to mine under the walls.
>The circumvallation is complete; now we shall turn toward building our contravallation lines.
>Launch the assault right now.
>Wait a turn.