However you answer this question or choose to even begin answering this question is conditioned heavily on the kind of story you want to tell. If your vision of this species and the role they played in your story was complete then the answer I think would either come naturally or not be entirely relevant to their portrayal. It is hard to believe that the way individuals of a specific species are addressed in your story would not be relevant to their portrayal, so thinking more about what kind of species they are and their role in the story might help the decision regarding what pronouns they should be referred to as come more naturally.
Regardless of what kind of creature they are, humans are the kind of creature where there are primarily 2 distinct genders and a great many qualities are attributed to the expression of each. These qualities range from emotional to behavioral to physical. This means that if, for example, the species was entirely genderless and yet every single individual was identital and looked remarkably like Dwayne "The Rock" Johnson, us humans would pretty much exclusively refer to them using the pronoun "he" regardless of what reproductive organs they had underneath their clothes and it would be weird to use "she" or "it" from our perspective regardless of if these were technically just as accurate.
What do they look like? How do they behave? Do they act aggressive or do they advocate peace? Are they competitive or cooperative? Are they emotional or are they stoic? Would they make better engineers or would they make better nurses? Are they tall or short? What do their voices sound like? How do they walk? How do they take care of their offspring? All of these, for better or for worse, might influence what pronouns us humans would naturally attribute to them.
That said, what pronouns they would use, when speaking English, MIGHT be a different matter. They would either use the same pronouns humans use, or not, depending on how they chose to approach translating their language to English. You say in a comment to your original question:
"Kelly was one of the oldest and wisest among us. Kelly was my mate.
Kelly supported me when... I wish to avenge Kelly's death." My first
thought was to just use the name, but it gets clunky to use it again
and again in a paragraph.
And you are correct: it would be awkward for a member of such a species to speak this way. The thing is, however, that they might have no reason to speak this way. Presumably, in order to speak English, some of the specific qualities of their language would carry over. For example: if none of their verbs are ever tensed, they might have a hard time using tense. This would be the same for pronouns. Say, for example, their language has no notion of a pronoun. The sentence:
Kelly was one of the oldest and wisest among us. Kelly was my mate.
Kelly supported me when... I wish to avenge Kelly's death.
MIGHT actually be naturally and literally translated by a member of this species to:
Kelly, among us both the oldest and the wisest, who my mate was, who I
supported by was when... will I wish by me be avenged.
But that might just be a literal translation. A more appropriate and less awkward translation might be something like:
Kelly, the oldest and the wisest among us, who was my mate, who supported me when... will be avenged.
If they don't have pronouns their sentences (in their natural language) would naturally be formed in a way that avoids pronouns, and so their attempts to translate their natural language into English would reflect this.
Again, this is all taking a rather realistic approach to the question, I.E. by answering the question of what pronouns we might realistically expect humans to use for them and how we might realistically expect them to translate their own language into English. You might find that the realistic answer does not fit in well with how you want the species to be portrayed, or what you want their role in the story to be. The reader will inevitably read how they speak and how they are referred to and create an image in their mind of the species that fits that portrayal, so ask yourself which pronouns best match the portrayal you are going for after asking yourself which kind of portrayal would be more appropriate to your story and why.