People ask this one a lot, especially if they are new to riding or thinking about getting their first bike. The short answer is that both work great on a pump track. The longer answer is that it depends on what you actually want to get out of it.
What a Pump Track Rewards
Pump tracks are all about generating speed without pedalling. You do it by pushing into bumps (rollers) and using your body weight through corners. This means the bike matters less than your timing and technique. A skilled rider on a cheap bike will always beat a nervous rider on an expensive one.
That said, the two bike types do feel very different on the same track.
BMX on a Pump Track
BMX bikes were basically made for this. They are light, stiff, and responsive. When you push into a roller, you feel it instantly. There is no suspension to soak things up, which sounds like a bad thing but is actually the point. Every bit of energy you put in comes back to you as speed.
The shorter wheelbase makes BMX bikes very flickable in tight corners. If the pump track has lots of small, fast rollers close together, a BMX will feel natural.
The downside is that BMX bikes have one gear and short cranks. If the track has longer straights or bigger rollers spaced further apart, you might run out of gear and feel like you are spinning without going anywhere.
MTB on a Pump Track
A hardtail mountain bike is a solid choice, especially if it is what you already own. You get more gears, longer cranks, and a more upright position. On bigger pump tracks with deeper rollers and wider berms, an MTB can carry a lot of speed and feels very planted.
The longer wheelbase means it takes a bit more commitment to get a good pump going, but once you do, it carries momentum well. If you ride trails as well as the pump track, an MTB lets you practise skills that transfer directly.
Full suspension MTBs are trickier because the rear suspension absorbs some of the pump energy. A hardtail is almost always better on a pump track than a full sus bike.
Which Should You Choose
If you want to get fast on a pump track specifically and plan to spend a lot of time there, a BMX or a 24-inch BMX cruiser is hard to beat. They are also cheaper to run and rarely need maintenance.
If you already have a hardtail MTB and want to use the pump track to improve your trail riding, just ride what you have. You will learn the same fundamental skills and the transfer is excellent.
If you are buying for a child, a BMX is usually the better starting point. Lower standover, lighter weight, and simpler to maintain.
One More Thing
A lot of dedicated pump track riders end up with both. They ride the BMX when they want to go fast and feel sharp, and they ride the MTB when they want a specific trail skill workout. There is no wrong answer here. The best bike is the one that gets you out there.