Cube maps (cubic format) are more regularly found in video game tech (perhaps your client has some experience in this field?).
Most consumer camera stitching tools, inc Insta360, will output into equirectangular format simply because it is the most widely used in the world of 360 photography software.
That said, when performing editing or analysis (e.g. object recognition). the cubemap format can be useful in the world of 360 photography.
Equirectangular images are stretched in the ‘latitude’ direction. It is the reason for a considerable amount of data redundancy near the poles. When downsizing an image in an editor, the effective texture resolution is decreased as expected - except near the poles. This can often cause problems when viewed later in panorama viewers.
An equirectangular projection is thus suitable for simulating only those environments where the distortions of the texture at the top and bottom of the sphere are negligible.
If you do want to scale, blur or sharpen a panorama it is better to convert it to cubemap first, a less distorted projection, make the edits, and convert it back to equirectangular format afterwards, if necessary, to avoid these issues.
Out of interest, what work are you proposing/doing for this client?