In my opinion, it's the "any" that sounds weird to me. I would probably say one of
"Either Varun or some other member of the group Schitzengiggles mentioned this in one of [his or her/their] videos."
"Either Varun or another member of the group Schitzengiggles mentioned this in one of [his or her/their] videos."
"Either Varun or a different member of the group Schitzengiggles mentioned this in one of [his or her/their] videos."
If you're going to say "one of" then you need to pluralize video to videos. Whether you use "his or her," "their," or just "his" depends on what you mean. You have to decide, for your sentence, who owns the videos. Is it Varun/the other member, or the collective Schitzengigggles? If you want to convey that the video is owned by an individual in the Schitzengiggles, then you ought to use "his or her." You could get away with just "his" in most informal contexts but be aware that it may not be received well in all cultural contexts unless it is known that all members of the Schitzengiggles are male. If the videos are considered to be owned by the collective Schitzengiggles, then you would use "their."
Optionally, you could also omit the word "group" as it doesn't add much value to the sentence and therefore only serves to make the sentence more clunky. Saying "member of the" is enough to indicate that the following noun is a collection. I would only leave "group" in the sentence if you have not yet introduced the Schitzengiggles or if there are multiple entities named "the Schitzengiggles" (i.e. an individual with that name and a group with that name) and "group" distinguishes to which you refer.