You're right. Your question cannot be answered definitively, at least not at this time and in our current state of understanding. The limits of human wisdom preclude the making of any categorical statement one way or other. We are left but to speculate, wonder, and believe. But belief itself is a product of having faith, which itself is defined as "being certain of that which we cannot see". [Heb 11:1 NIV]
We must therefore depend, if we want to have faith, on revelation. If you are oriented to Christianity, the source of that revelation is the Bible. Evidently however the Bible does not want us to know things which we cannot see, because if we knew for certain it would by logic cancel out the need for faith.
The Bible , is either deliberately or otherwise, too vague, or ambiguous, or even subject to a myriad different interpretive conclusions, that one will find onself miserably locked into a maze of conflicting values if one ponders a question such as this. The two parts of the Bible are already locked into their own separate theolgcal matrix and the condition of animal life is left unstated. The OT is concerned about the relationship of the nation of Israel and its descendants to their God Yahweh, while the NT deals with the issue of sin and reconcilliation back to God through the blood of Christ. No mention of Tabby or Trigger anywhere in either hemisphere of the Bible.
But there are tantalizing clues. For instance we know that animals are CALLED souls, but do they also HAVE souls? Dunno. However there is a text at Prov 12:10, where Freddy Franz translated the NWT to read:
"The righteous man is caring for the soul OF his domestic animal"
The genetive here evidently indicates possession, so that the "of" implies that domestic beasts, at least, appear to HAVE a soul. However, as I said, this is a minefield of interpretive impulses and is subject to endless debate.
Does "soul" here mean "life" only? According to the NASV Bible the answer is Yes. Can the word "soul" here [Hebrew "Nephesh"] simply mean the physical needs of the animal? According to the NIV, the answer is Yes. [The righteous man tends to the NEEDS of his animal - NIV]
And so on.
What then do we believe? That is what the root of our humanity is, and not what we know or want to know. Personally, as a dyed-in-the-wool Baptist, I believe that only domestic pets who are smarter than their owners will live with them forever in the eventual heavenly Paradise.
Which means that Moogy my cat is a shoo in. She's so smart that she has mastered the "fetch" command. When she wants something she gets me to fetch it.