Understand that _nothing_ in biology is _random_ what this means is that choosing your gene needs to be regulated by _something_. Temperature, light, pressure, amount of a transcription factor.
Next, once a genetic change is made, say by using Lox/Cre or CRISPR, there is no going back(or it is very very complicated) so the "randomness" would only be good for one generation. i.e. you transform a bacteria with the original plasmid and it will randomly be a color but each generation after that will be the same color.
However, if that is all you want then the simplest approach would probably best be done by creating a library of plasmids with promoters that allows a constitutively active transcription factor to activate only one of the genes at a time in the cell. Then transforming your bacteria with all of them and each will contain a random gene that they activate for one generation or maybe multiples to create different combinations of pigments.
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There are different versions of brainbow and, if I recall correctly, there is at least one that involves inversions of DNA not excisions (so there is no loss of information). Cutting it short , you need to control the cre expression with a promoter that only activates during division. This way you force to rearrange DNA in the division events and the color should stay the same for as long as the cell doesnt divide.
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