Stuff growing under a liner is occasionally a problem, usually in the form of dark stains that can't be removed because they come from underneath. But, I've never known of such stains being more than a cosmetic problem.
I don't know that putting new sand mix down will kill anything, and the effect from bleach is likely to be temporary. If you want to experiment, you could make up a copper sulfate + borax mixture and spray with that: it would be longer lasting. You'd probably want to start with a strong water and copper sulfate mixture, and add borax till you get a neutral pH . . . and then spray that.
BUT, that would be an experiment -- I've never done it -- and there might be some weird reaction from it that would damage the liner. So, no guarantees. But such a mixture would probably be a pretty strong fungus & algae inhibitor.
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