In US law, under 17 USC 115 A person can obtain a "cover license", also known as a "mechanical license", to perform and record a version of a musical work protected by copyright.
This license is compulsory, the copyright owner has no right to refuse. One simply sends a notice to the appropriate collecting society specifying the work to be recorded, the person or entity doing the recording, the release date, and various other details. One also pays the statutory royalty rate, which I believe is currently 9.1 cents per composition of 5 minutes or less (per copy) or 1.75 cents per minute of recording (per copy). This payment entitles one to create an arrangement or "cover version" of the piece, perform that version, record the performance, make and distribute copies of that recording, all lawfully.
There are some limits on this license. It is on;y available if the composition has already been performed and recorded, and the recordings distributed, by permission of the copyright owner. It is only available for retail distribution of the recordings. If the melody or lyrics are changed a different process needs to be followed, where the owner can decline permission.The compulsory license also does not cover a synchronization license, which is what is needed or a music video. This must be negotiated individually. A cover license also does not cover public performances broadcast or in front of an audience. It only covers making recordings.
See "Mechanical Licensing & You: What You Need to Know Before Recording Your School's Performances" for a description of the process
Assuming that Halocene went though this process properly, they have a license to distribute their cover version, and a copyright on their arrangement. Or Halocene might have negotiated a broader license, one permitting public performances or other things, I have no way to know what deal they made.
Masked Singer's Rights
Australia has, I understand, a similar legal process to license a cover version of a song. The "Masked Singer" might have gone through a similar process, but that would not cover public performances, nor te details of Halocene's arrangement. It is hard to see how rights to the specific arrangement would ha been lawfully transferred.
Copyright infringement cases are often very fact-specific, an I can't know what the details oif the facts are. But it sounds as if they might have a case.