How I play quiet bottom notes.
It is possible to play quietly by squashing the reed with your embouchure to make a smaller aperture for the air to go through, but this makes the note sharper and it is not always possible to keep the pitch correct at the same time.
Here is a method that I find better:
The secret is to realise that notes tend to stick out when the tone is bright but sound quieter when the tone is dull. So now the problem is how to change the tone. You probably play with your lips curled over your teeth and with your lips one on top of the other. This is not the only way to form an embouchure!
Reach forward with your top lip and place it further up the reed than you usually do, then uncurl it a little (perhaps by pushing the instrument away from you) so that more of your lip is in contact with the reed. Now bring your bottom lip nearer the tip of the reed. (If your method of holding the bassoon is very rigid you may find this awkward.) When I do this it has the effect of dulling the sound and enabling me to blow harder for the same dynamic. You have more control over the pitch with your lips and can play flatter if necessary.
Needless to say that a good reed will help! To get notes to start quietly you need to check that the tip of your reed is not too thick. At 1 mm from the tip I aim for about 30/100 mm in the centre and 20/100 mm towards the edges. I also run a piece of Dutch rush at an angle of 45° across the tip of both sides (you can of course use wet and dry glasspaper). Beware of doing this too much or you may cut your tongue on the reed.