I think one thing to think about when it comes to spaces is that it's kind of hard to regulate.
For instance, you could have a name like Britney Spears with a space in between.
But what about B ritney Spears?
Or Britney Spears with two spaces in between?
Or Britney Spears with a space at the beginning and end?
They're all virtually the same name since spaces really don't add/mean anything and that kind of takes away from the uniqueness of names.
You could try to impose limitations but where would you draw the line?
Just my two cents.
But I understand your frustration with naming pets.
Personally, I turn to other languages for inspiration.
Or if I want to name a pet after a fictional character or something I name them after something the character does/owns/says.
For example, I wanted a tribute pet to Amon from The Legend of Korra.
I decided to name them IAmtheSolution as that's something he says in the show.
It may not be ideal and may not be your cup of tea but just a thought.