My husband is an attorney and he says...

Judges are very territorial and may have rules about their court and others do not. Unless the rules are very egregious, and illegal, there is not much anyone can do.

I saw a trial where not ONE spectator was allowed to cry or make a sound of any kind!
I saw one trial, where the mother of a victim was NOT allowed to show any emotion during her testimony. This was on CourtTV/InSessions, and the attorneys commented on it.
Some courts will not allow any spectator to read during the trial.
(this happened when I was called to jury duty once). At that trial no gum chewing was allowed either for spectators!

I am not saying you are in the wrong. I am saying that the judge may be pretty powerful. Some are really strict and others let people whisper during trials...it swings both ways.

If this judge was elected...a public outcry in the media may be helpful, and watch for the re-election and bring up again.
Going thru the judicial tenure committee format may only be helpful if there are other complaints against him, and takes time.

Some judges will limit anything that distracts the jury from listening to the case. ANYTHING. Some will not allow cameras, and others will. This is why you were allowed your dog in the first trial, and ran into a problem during the second.

Alaska has some unusual laws too. Did you know birth and death records (which are public in other states) are kept sealed in Alaska for 100yrs (for birth) and 50 yrs for death/marriage/divorce) !??? (for example)
http://www.hss.state.ak.us/dph/bvs/

Is there an association for therapy/medical dogs in your state?
I'd join them and as a larger body, they may be able to shed more light on your plight and help you too.

These are just my observations.
If it were me in your shoes, I'd get some meds from the doctor to help me attend a trial like this. It has to be very difficult and painful, and I couldn't do it without medical help.