"I do not keep anything except for my dog."
Translation:אני לא שומר שום דבר מלבד הכלב שלי.
The English sentence is ambiguous. It could mean "My dog is the only thing I keep." It's this meaning that's translated into Hebrew here. For this meaning, the English could use "except" instead of "except for", and "except for" means nothing more than "except". But it could also mean "The only things I keep are things that I use to take care of my dog", in which case I think that בשביל would be part of the translation, and "except for" would be shorthand for "except things that I use for".
It's not wrong to add את here as far as I can tell, i.e. אני לא שומר שום דבר מלבד את הכלב שלי. However, unlike most sentences with a direct definite object, it does sound better to drop the את here, at least to my ears.
There isn't an exception for negative sentences - "I don't see my dog" = אני לא רואה את הכלב שלי. The 'exception' is probably just for sentences with 'except' like - "I don't see anything except for my dog" = אני לא רואה כלום חוץ מהכלב שלי/אני לא רואה כלום חוץ מאת הכלב שלי (but it's more common to drop the את here).
I agree with radagastthebrown that adding את is possible, and less common/natural. I think the reason is that את is used when the direct object is a (well) direct complement of the verb. Here the direct complement is חוץ מ, and הכלב שלי is direct complement of the חוץ מ, so in the syntax tree it's one level separated from the verb.
In this light, it's maybe surprising that את can be used at all... Disclaimer: this explanation is deep linguistics territory, and I'm just an amateur linguist...
Actually, "direct complement" is probably not the conventional term linguists use. Are you familiar with syntax trees? https://essentialsoflinguistics.pressbooks.com/chapter/8-2-tree-diagrams/ is a crash intro. My conjecture is that את is used if the definite thing is the sibling node to the verb. Here the sibling node is rooted at חוץ, and the definite thing (הכלב שלי) is one level down inside the subtree of חוץ. Does it help? /-:
You're right. "Except" and "except for" can be used interchangeably as prepositions -
and מלבד is a correct translation for both.