Hi , I'm James DVM MRCVS and would like to help you with your question
how is Bertie otherwise ?
is he eating and drinking ok?
Has he lost any body weight over the past months ?
Is he showing any strange behaviors ?
One of the more possible things which come into mind is indeed core temperature.
You see, as Bertie has whippet genes (meaning not much insulation between skin and bones) as well as him getting older which means that the natural mechanisms of thermo -regulations are not what they used to be when he was younger . all that brings me firstly to believe that he is cold and hence "clicking his teeth" . When we sleep our core temperature drops which is why we try to keep extra warm when we sleep , same goes for dog .
it could very well be that Bertie is suffering from a low level hypothermia which results in shacking and teeth clicking - a natural response of the body to try and produce more heat .
Other options are neurological or muscle related disease of the jaw but that would manifest in other signs as well like problems eating and , excessive salivation and other more general symptoms involving the whole body.
AS a first step try to increase the heat in his environment as well as cover him or dress him with warm tops (as funny as it may sound) . I would suggest as well to run a general blood test and perhaps test his thyroid gland levels as well - at times this gland grows tired and as a result the rate of metabolism drops which as well means the drop in core temperature .
I don't think this symptoms is a result of anxiety so I doubt that rescue medications such as Kalms will help , more so that it is not licensed for dogs so I will strongly urge against it .
I hope this helps .
If you have any further questions , please let me kknow.
All the best
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You didn't mention that Bertie is having kidney problems. When was the last time he had his renal blood values checked? We might be dealing with a systemic problem which again will indicate a full blood profile to be done.
Another thing you can do as just to test would be to buy a thermometer and check his rectal temperature during the day. At times when he is fine and at times when he is clicking his teeth.
Come back to me with the results once you have them