... and the end-cap should slide down

There are two catches on the sides holding the two halves together, but rather than opening these ...

... it's easier to insert a thumbnail in the join to open up the first catch, then a second nail on the other corner to open up that side. That gets it partly open, but another finger-nail is needed further along the sides ...

... to fully click the two halves apart.

Flick out the battery with a small screwdriver

Clean the contacts and the battery (CR2032) with a clean, dry cloth. Insert the new battery positive end (+) down such that the sides touching the contacts (arrowed) go down into the hole first, then press the opposite down. Don't touch the battery with your fingers.

Clean the battery contact pads on the printed circuit (arrowed) with a clean cloth, make sure the printed circuit is correctly located by the side tabs in the case, then press the battery half of the case down onto the printed circuit half.

Test the fob before refitting the end-cap.

The old case looking like something from a horror film

The key-ring that came with the car, ...

... it has a swivelling centre section. I've not seen one like this anywhere else, including in this page of hundreds on Google.