Yes. Individual apps don't actually handle this. The Bluetooth device automatically sets up a messaging protocol with your phone and it will receive the messages completely separate from the SMS apps on your phone. The Bluetooth device acts in a similar way to any SMS app on your phone.
If they don't seem to be coming through, you could try re-pairing the Bluetooth device, restarting the phone, or reinstalling Pulse SMS. All of these could wipe out any system-level integrations. Pulse SMS does not have, and will not have any configuration around Bluetooth devices. Individual apps do not manage this connection.