As Audioguru advises, it is fairly easy to interface the device with a relay, but a bit of electronics knowledge is needed. If the output is 2.1 volts DC, then I would take one approach, but if it is 2.1 V of ultrasonic AC, then I would use another. I'm assuming that it is 2.1V DC that turns on an ultrasonic sender.
I also assume that the output of the bark sensor is only on during the actual barking, not when the barking stops. In this case, you will probably want to add a circuit to extend the operating time of the sprinkler. How long would you want the sprinkler to run for? A timer circuit would be needed that is triggered by the bark sensor and then turns on the transistor switch that audioguru suggests, for a certain wait time. Alternatively, you could simply use a "time delay relay" which has the timing function built-in. This might be more convenient, but the time delay relay will cost a few dollars more than a basic relay.
Let us know what the fellows in the shop suggest. Another alternative is to have an electronics student built this up for you. It is simple enough to be within the capability of any second or third year college student in engineering or electronics tech. I would be happy to build something for you, but being in Canada makes the shipping annoyingly expensive. PM me if you want to follow up.
I'm also curious to know whether the Bark Smart seemed to do anything or not. Could you tell if it was detecting barking. Is there a lamp that lights up when it is sending?