Haleyjay, I am sorry you are going through this.
Based on your other threads, this needs to be taken seriously. Yes, it is likely that the person who denounced you is acting maliciously but you need to set things up so that you have proof that any future accusations are baseless because unfortunately you, or more correctly your dogs, are vulnerable here.
Nasty neighbors know this, they know they are hitting where it hurts most, they are using the things you love, the dogs, in order to get to you. Such people get away with it because most of us will first protect our dogs. This is a very real, very ugly side of life in Switzerland.
How long are the dogs left for? Can you get a sitter in during those times? Failing that, can you get a recording to show when you left, when you returned, what the dogs were doing while you were gone? When it comes to your word against another person's anything can happen. You would be best off if you had irrefutable proof that the accusations are baseless and that the dogs are fine when left alone.
I'd also consider seeking expert help. You might consider contacting Tier Im Recht - and if they could not advise you perhaps they might be able to steer you to someone (likely a lawyer) who can.
And - as IIRC was mentioned in the other thread, gather testimonials from 'authority figures' who know you and know how you treat your dogs. Your SKN trainer is a good person here.
What happened when you spoke last with the landlord over previous harrassment? Did the landlord take any steps to help you, or offer any advice?
Insist on a copy of the animal welfare report. You need to know what the findings were, and you need to address any points brought up. Hopefully the report will clear you completely, even so you also need to keep a copy on hand, should further complaints/harrassment/ugliness occur.
Was the conversation with Animal Welfare conducted in English, or in German? If the latter, is your German fluent? If not, in the future insist on either English or ask to reschedule for another time so that you have someone there to translate for you.
Honestly - were it me I would consider moving. If the nasty neighbor has gone so far as to call Tierschutz the person is either seriously unhinged or determined to make trouble.
And were it me, I would use this incident of yet another example of harassment making your living there untenable, a reason to ask the landlord to let you out of the lease early, as soon as you have found someplace else.
Thumbs pressed that this clears up without too many more tears.
Wishing you and the dogs all the best.