It turned out that this was simpler than I've thought. In Plan Configuration, in the Repositories tab, on each repository, under Advanced, there is an Include / exclude files input where you can Customise what files Bamboo uses to detect changes. By adding a regular expression there, I got everything solved and working as expected.
Bamboo pattern matching reference: https://confluence.atlassian.com/display/BAMBOO/Pattern+matching+reference