The phenomenon of "shaky" notes may take several forms and be due to quite differing causes.
It seems as if your diagnosis is based on embouchure and certainly this is one of the most likely factors.
If the bocal needs attention this should be done anyway - as with any mechanical adjustments or corrections.
Does the problem persist if the student (using own reed) plays on another bassoon?
Once you have eliminated the instrument as a possible cause, then work on breath support and embouchure.
Playing too near the tip allows over-control of the reed by the jaw muscles and creates a rattly tone, usually flat pitch, and a stifling of the overtones; generally all bad features.
A low support pressure even for pp playing can create a wobbly vibrato effect and (if the vocal folds become involved) a hesitant "catch" in the sound.
Personally I avoid trying to quieten my students down too early in their playing experience. My theory is that if a young player is found to be too loud (in a band or orchestra setting they are often just out of tune (which is too loud at any level) and with encouragement to play out with good support and relaxed embouchure, once in tune are found (even at louder levels) to be quite welcome and in balance. A stiff reed obviously will be raucous and should be adjusted.
A good measure is to have the student play notes from open F down to low C - lips on the binding using only supportive breath pressure to keep in tune. A poco forte level should be achieved which is in tune. Most junior playing should be sustained like this for at least the first year or two then gradually bring in the quieter dynamics. A student can also learn good dynamic control on individual notes by playing a sustained mid range note and tapering it down by simultaneous lowering of breath pressure with a balancing gradual increase in the "bite".
I compare it to an aircraft landing - less forward thrust (breath pressure) - more lift (bite). My students learn quite early on to make "good landings".
Hope this advice progresses things but feel free to follow up.
Christchurch New Zealand
Bassoonist / Teacher / Conductor