Hello Chris, excuse my sarcastic nature. Grumpy old fart syndrome.
The correct answer is this. Fabric used over RIGID fiberglass insulation SHOULD be "acoustically transparent". At least where it doesn't IMPEDE the movement of air molecules. This is why audio monitors in studios are NOT covered with grill cloth, as ALL fabric will impede the soundwave to some extent. Some, moreso than others. Some have even been tested. Like Guilford of Maine fabrics, which are specified in many studios. Some homerecording people use muslin, or even burlap, although I don't recommend it unless you spray it with a fire retardant product such as 3M makes.
The second part is BATT refers to the TYPE of insulation used in the cavities of walls as THERMAL insulation. This type dampens the vibration in isolation wall assemblys, is cheaper to use in that application than RIGID fiberglass. and performs quite well for this task. As a cavity is an air gap between two LEAVES of a MASS AIR MASS assembly, the BATT insulation also performs absorption tasks, although not to the extent that RIGID FIBERGLASS does, as the density of BATT type insulation is much less.
For BASS traps, and general absorption panels, the type product usually used is OWENS CORNING 703, or 705 RIGID FIBERGLASS PANELS
depending on what you need in absorption. 703 is UNFACED, which means it does not have a covering on either face. Therefore you need to use a fabric to keep from brushing against it. Plus it looks better.

And yes, it has loose fibers if you rub it, it will release these into the air. However, rigid fiberglass fibers are bound together by a heat process(I believe) or a binding agent. No matter, it doesn't act the same as BATT type insulation in that regard. If you want complete isolation from the fibers, then if used in frames, simply staple a layer of poly(3 mil should work) over it, as this will also
give reflection qualities at higher frequencys, and I believe lower the bandwidth a bit, but don't take my word on that. I also think a
705 comes with aluminum foil on one face which does the same thing as I just explained. These panels come in thickness's from 1/2" to 6". And in panels 2'x4', and maybe in 4'x8', but I've never encountered them that large.
Remember, the thicker the panel, the lower the bandwidth of absorption.
Also, when mounting, do NOT mount directly to the wall. Leave
1/2" to 2 or 3" airgap bbehind them, depending on the thickness used. This lowers the bandwidth also. And actually, Rigid fiberglass works really well if simply placed ACROSS the intersection of walls, walls and ceilings, and to some, even the floor/wall intersections. That is where bass frequencys form.
Well, hope this helps clarify some things. And as usual, here is my disclaimer. I am NOT an expert, and generally just try to pass information on to uninformed people. Some may dispute what I say in some form or other.
fitZ