This will be easy. I would recommend using a smartthings brand button instead of the flic because it’s less expensive and works directly with smartthings. Otherwise you have to use Ifttt to connect the flic to smartthings and again, the flic costs more. (I like flic and use some myself, I just don’t think it’s the best option for this use case.)
Then you would set up automations for the following:
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when the ST button is pressed ( or any other device that smartthings can recognize), unlock the August lock and turn on a virtual switch/sensor that starts an Alexa routine (not a smart things routine)
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The Alexa routine can have any specific echo device or multiple echo devices say any specific custom phrase that you want it to say. So that’s the easiest.
The community FAQ explains the exact steps you need to do to set up The virtual device.