Can I use a different magnet to trigger the BT510 sensor magnet switch?

Technically any magnet with sufficient field strength and proper field orientation can be used.
The magnet sensor used in the BT510 is a magnetoresistive sensor and hence the sensitive axis is in the sensor plane (other than for hall sensors where the sensitive axis is perpendicular to the sensor plane). Below picture indicates the sensitive axis. The sensor is NOT sensitive to the direction of the field!

MR Magnet Sensor

More information about sensitivity and orientation recommendations from the sensor chip perspective is in the SM351LT data sheet. The below picture summarizes the most important aspects. It also shows the orientation of the sensor chip relative to the BT510 enclosure. The magnet coming with the BT510 is a rod-style magnet with axial magnetization and hence the field lines go from the short sides (top/bottom) of the magnet (more or less parallel) along the long sides of the magnet like indicated in the “magnet movement” section of the picture. This orientation ensures the field lines penetrating the sensor chip mainly along its sensitive axis. One can also see that rotating the magnet by 90° would lead to the field lines penetrating the sensor chip perpendicular to the sensitive axis and hence not generate any signal.

BT510 MR Sensor and Magnet Orientation

When selecting a different magnet than the standard BT510 magnet both field strength and orientation needs to be taken into account. A magnet with higher field strength can cover a larger operating distance to trigger the magnet switch. 

There exist a lot of standard round magnets on the market often used to stick/fix something onto metal surfaces and hence the magnetization is not critical and works for any magnetization type (like axial, vertical, diametrical…). However, often the magnetization type is not even specified for a particular magnet and cheap ones (like used for white boards) sometimes even have several magnetic poles in parallel over the area of the magnet. This actually helps for the common use-case of sticking well to metal surfaces but makes them quite unusable for reliably and reproducibly triggering a sensor chip.

How to best orientate your round magnet to get stable sensor near/far readings would depend on how the magnet is being magnetized and how the magnetic field lines would penetrate the sensor chip when applied. If the magnetization is unknown a magnetic field viewer (aka flux detector/viewer) can be used to "see" the magnetic poles and get the orientation of field lines.
The general advice would be to either use a rod-style magnet with axial magnetization in the above shown orientation or a round magnet with known magnetization type and then positioned in a way so that the field lines penetrate the sensor in its most sensitive axis. To ensure this, the magnetization of the magnet must be known before mounting.