Figured this one out. Basically to get the image once you've got the filename you need to do another GET to /chart?filename=<your-filename.png>.
So, if you use the following to get the chart location:
$ curl -XGET --user name:password "https://my.bamboo.com/rest/api/latest/chart?reportKey=com.atlassian.bamboo.plugin.system.reports:averageTimeToFix&buildKeys=MY-BUILD&dateFilter=LAST_30_DAYS"
<?xml version="1.0" encoding="UTF-8" standalone="yes"?>
Then you can get the chart image by doing the following:
$ curl -XGET --user name:password "https://my.bamboo.com/chart?filename=jfreechart-onetime-583392983747583030152.png" > ~/Desktop/chart.png
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