Not sure if the vegetable dye would work,but it might be staining and messy.We have soy paint here in the US,but as mentioned it would be easy to chew off.If there is a nontoxic water based coating that works on plastic without primer,that could be good to look into,but other than that I think it would be safest to use colored paper and wrap it with packaging tape on the outside of the cage only where it can't be chewed.
Another idea is duct tape,but only on the outside of the cage of course with the same precautions as paper!Anything along the lines of duct tape would be okay as long as the hamster does not have access to the tape or paper or whatever you end up using
Just thought of this as well,but wrapping paper would be another idea as well!