Bryan,
I believe the problem might be what is being sent to the CL_HTTP_UTILITY=>SET_REQUEST_URI. The SM59 destination is configured such that when you create your HTTP client, you're already working with an URL of secure.smartwaregroup.com/Bigfoot/OURCOMPANY. So when you set the URI with /Bigfoot/OURCOMPANY/api/methodname, you are double dipping and the web service is attempting to reach secure.smartwaregroup.com/Bigfoot/OURCOMPANY/Bigfoot/OURCOMPANY/api/methodname. I think if you just send over the '/api/methodname' it should work. If not, I can provide another example of how I got this to work on my end by using CREATE_BY_DESTINATION/
Thanks,
Brian