I used a 15mm tall, 2.5" 4TB Samsung P3 External HDD, which I removed from its enclosure. I believe the PS4 only fits 12.5 mm tall, 2.5" HDDs, and I knew going into this 15mm wasn't going to cut it (figuratively), so I would have to do so instead (literally). I cut the shielding which envelopes the caddy.
As I have a CUH-1200 model, the caddy itself is actually top-mounted on the HDD (believe the 1000/1100 models mount to the bottom). Because of this, the screw mountings won't support a 15mm HDD. Additionally, there's a slight height gap for airflow beneath the bottom of the HDD (as it's typically suspended). As such, I inverted the caddy, using it as a tray, set the new HDD inside it (much like the 1000/1100 models) and aligned it with the SATA connector.
A long-term approach will involve drilling holes to properly mount the HDD with screws, but, for now, I taped the HDD to the tray, and the tray to the console. It's somewhat of a jury-rig, but it's plenty secure.
The casing itself took a bit of effort to reattach, and is ever so slightly bowed outward, though I've put eyes-on, and it's largely imperceptible. Everything is now snuggly fit, which will likely increase heat retention, but not to a degree where it will be damaging. The HDD is only 5400 rotations/min (same as stock Drive), and the 1200 model PS4 runs cooler by design, making any additional heat build-up negatable.