First off, parsing song lyrics is in the same vein as parsing poetry. Sometimes there is deliberate ambiguity, and no one knows what the true meaning is except for the writer.
I acknowledge that your question is about grammar and not necessarily meaning but song lyrics, like poetry, play fast and loose with prescriptive grammar.
That being said, I think the key to understanding the line "その時は頼りない爪でひっかいて" in this case is by looking at the line that comes after:
the use of "あげる" here suggests the direction of the action in that clause. "あげる" always speaks to the subject doing something for someone else. So the subject is doing the protecting, and we can assume that the subject is "私" because otherwise, the speaker would be saying someone would be protecting someone else. You can't use "あげる" to say that you receive something only to say that you do something for someone or that someone else does something for a third party.
So the subject of the first line is "We/Us" because of the verb ending in there, and then we get to "守ってあげる" with what we have to assume is the same speaker (there is nothing to suggest otherwise).
are both grammatically valid segments, it would just be odd from a Japanese sentence for the subject to shift like that without any indication.
In English, it would be fine because we would explicitly include some indication as to who is doing the action:
I would scratch it with my claws
scratch it with your claws
but in Japanese, it's just not necessary; in this case, though, the fact that it isn't there means we would have to rely on some other context for that info, and the surrounding context suggests that the speaker is the doer of the action and that the "て" is acting as a conjuction. So taking some creative liberties (which I think is necessary when deciphering poetry and prose), and focusing on localization, I would interpret the entire thing as:
Let's keep walking toward the ocean
Where we might find great beasts
Then I would scratch them with my unreliable claws
To protect you
I was wondering about the next two lines, and checking the original lyrics, I see where they are separated. I think this represents a separate thought.
I would interpret this part as (again, taking some creative license and focusing on localization):
What could I say
that could make me stronger
What could be said
that could make us stronger
In this context, because we are dealing with song lyrics and because of the nature of Japanese, both are valid interpretations. We could debate, but there is no way of telling from the rest of the lyrics. The writer may not have even had anything definite in mind because the language naturally allows for this kind of ambiguity.
I suppose my conclusion here is that these things are usually clear from context. Generally speaking, you can sometimes use the subject or the doer of the action to determine between an imperative or conjunctive "て”, but the subject isn't always necessary in Japanese. In that case, you just have to derive from the surrounding context or accept the duplicity/ambiguity in the interpretation because that is completely valid in Japanese.
This is a nice little article on ambiguity or 曖昧 in Japanese. This phenomenon is well researched and documented, so you'll be able to find a good deal of information on it. I think understanding this concept is essential if you truly want to approach native-level fluency. Japanese people often talk about foreigners not being able to "read the air" 「空気読めない」and part of that has to do with this concept.
I didn't mention the trailing "て” here, which I think is an integral part of my interpretation.
Also see @bulgar69's interpretation. It kinda throws ambiguity back into the mix for me where "ひっかいて” is concerned. But his interpretation (which uses the imperative) makes more sense poetically to me! So I'd vote up his answer over mine if he posts one.