Dates (and, for that matter, measuring time) comes about very, very, very early in a civilization. A fairly complete date and time system would have been developed by your cats long (eons) before your Surface Hunters came to be. So what do we have to work with? The simplest thing I can think of is...
The Circadian Rhythm
Your cats need to sleep and I assume (being based in Philadelphia) that they originally evolved on the surface. This means your cats need to sleep. Cats today sleep on average 12–16 hours a day. Let's use the average for convenience: 13 hours. That means they're up and about 11 hours a day on average.
And lo, as the ancients wept through the darkness in search of light they called upon Felidae for help and wisdom. And behold Felidae did bless the ancients with wisdom, for she called the time of rest "death" and the time of searching "light". And Felidae did watch over her children, over life and over death, and behold she called this the first Day, and it was good.
So your cats are working with what we humans might call a 26-hour day, but for the moment, all we know is that they have a way of tracking days: the time spent awake plus the time spent sleeping is a day. Is it accurate? Heck no! But while sunrise-to-sunrise might be more accurate, it's not actually precise because it changes depending on your latitude, the time of the year, and your moment in history because the length of day has changed compared to days past. So, for our burgeoning intelligent cats, we now have a Day.
The Value of Religion and Philosophy
Now, while historically we might be doing this in order, religiously we're putting the cart before the horse. The overwhelming influence of religion and philosophy on calendars and time cannot be overstated.
In the beginning was the void. The void was dark and without form. Felidae did look upon the darkness and new that without life the darkness had no purpose. And Felidae wept for without purpose all is truly nothing. And her tears fell upon the void and washed through the void and separated the void from the void and filling the void and Felidae found hope.
And Felidae did breathe upon the void that was filled with her tears and the water did recede and there was space for purpose and Felidae once again found hope.
And Felidae passed her paw over the space and the space did embrace Felidae and the touch of Felidae did cause the blanket of life to rest upon the space: the mushroom and the lichen, the moss and the short grass. And Felidae smiled that purpose was growing and again found hope.
And Felidae did wish for purpose to be fulfilled and so created life that would fill that purpose: the insects withing the space and the insects within the void, the mouse and the bat, the burrower and the crawler and that which slithered upon the grass and Felidae once again found hope.
But purpose was not yet fulfilled and Felidae did weep once more. Her tears fell upon her cheeks and upon her paws and upon her tail as she curled upon and around the void. And as Felidae did embrace the void and the void, Felidae, she did close her eyes and she did bless the void and her Children, the Ancients, were born. And Felidae did breath a great sigh and her Children did know life and did know purpose and began their journey through the space and through the void. And Felidae did find hope again.
—Insert that first verse here—
And Felidae did bless the day her Children were born and did command her Children to rest upon the fifth Day that they may remember the blessings of Felidae.
Now let's look at the life around your Cats
OK! Now we have days and weeks. For months let's look at the life cycle of the Kentucky Cave Beetle. As it happens, their life span is approximately one terrestrial month. Rounding a few numbers and playing with the averages, what we have is a month of 28 days, which is as inconveniently divided into 5 as our month is into 7, but it does give you a "season" of 5 months.
- Day = one wake + one sleep cycle.
- Week = 5 days
- Month = 28 days
- Season = 5 months
As for hours, let's ignore the dew claw
Cats have 4 claws on each paw if we ignore the dew claw. So, 8 hours of life and 8 hours of death. 16 hour days.
And the coolest thing about this is that this dating system conflicts enormously with what your Surface Hunters find outside. They'd see the moon. They'd see sunrise and sunset. And those periods of time would be completely at odds with their own dating system — with the exception of months! Cool!