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Either a whisper HOB or submersible filter (e.g. Elite) will really be effectively silent if you set them up right. I am sensitive to sound and can use either.
Normally I'd say get a quiet HOB or internal filter, but the actual answer to your question is yes. You can get a quiet air pump to run a sponge filter or go without filter entirely if you're willing to do once-twice weekly 50% water changes. RCS are so low bioload.... just don't get froggy with it and add a school of Pacu. lol
I was thinking about trying to put it by the window and let the riccia/hygro do its job, but the problem is I don't know if that'll be enough. I'm not totally sure how many I have, but I'd guess around 15 adults and 20+ babies?
I wouldn't go without a filter. I raise RCS for my lfs. RCS are extremely sensitive to any ammonia and will die even at what is considered "non-toxic" levels.