Execute terminal command "rsync" from BetterTouchTool
According to the format of the answer to that question, you'd want to specify the full path to the rsync program. I looked around a bit and found my rsync executable inside the folder /usr/bin/.
If your rsync is in the same place (it should be), you should modify your command in BetterTouchTool to:

/usr/bin/rsync -r /Users/owendostie/Dropbox/Ohouse /Applications/MAMP/htdocs/Ohouse


Unfortunately I don't use BTT so I can't test this, but that is how the other answer solved the issue.