Death by curse is not death by sword
Assuming the link in the comments is the correct version of the homebrew, the death is supposed to be some forgotten curse. But the details of the curse are up to the DM. So build out the curse!
First off, the description states:
"Felis live up to 135 years of age. It is whispered that they possess nine lives and, by some forgotten curse, they burn through each life every 15 years..."
So quick math, 9 lives times 15 years, equal their lifespan of 135 years. So the curse also prevents them from living past 135 years.
They live a normal life for 15 years, then the curse takes affect and they die. For the first eight times, they come back. On the ninth death, it is forever.
So the curse may not be they can automatically come back eight times, but that they can only live in 15 year spurts.
If that is the case, then dying early does not mean they can come back. It just means they didn't reach the 15 year span to allow the curse to take hold. So death still means death, unless they die via the curse.
Given that, if they can only resurrect if killed by the curse, then death still has an impact on a day-to-day basis.
Next, since it's part of the curse, the Felis would know ahead of time that their time is up. You can incorporate that into the story.
When it is their time, they need to make preparations.
They don't want to just die in the middle of the forest where they can be eaten. Remember, they "travel alone" and "rarely forming strong attachments to non-Felis", so they can't depend on someone else to watch over the body until the next life kicks in.
Add some rituals to be perform so that they come back "safely". There isn't a time-line for how long it takes, it might be an hour, it might be a day. It would suck to rent a room at an inn, die, and have the cleaning staff find you dead. No telling what would happen with the body.
Also incorporate a sense of dread when the time approaches. It doesn't have to be "just like falling asleep." There could be discomfort as the first sign, that gets worse as the time grows near. Leading up to weakness, fatigue, and/or exhaustion.
5e already makes it pretty easy to survive death, don't make it easier
Having a race that gets 8 free self-revivify spells is a little OP. This doesn't even kick in until after any "if brought to 0 you have 1 hp instead" feature, and after best-3-out-of-5 death saving throws.
You say it's part of the character backstory, but don't give specifics, so I really can't speak to that. But I would recommend editing the backstory to incorporating the every-15-year curse, and not the die-8-times version if at all possible.