Solution
To send a triggered (asynchronous) message while periodic (scheduled) messages are active, use the LabVIEW example: Example 06 – Transmit Scheduled and Dynamic Messages. Path: C:\Program Files\NI\LVAddons\ballardinstr32\1.6.0.3\examples\Astronics Ballard Technology\BTI429Lib\Example 06 - Transmit Scheduled and Dynamic Messages

You must install the
Astronics Ballard Avionics Driver to access the example and the BTI Editor. Use
the BTI XML Editor to define ARINC 429 channels, labels, and transmission mode (scheduled vs asynchronous/triggered). Then, in LabVIEW, open the BTI device, load the project/XML, start the schedule, and issue a trigger to send the asynchronous message.