Is it a violation to run low voltage wire in the same bridle ring as fire alarm cable?
The short answer is that sharing a bridle ring with another power limited circuit is usually not a code violation, but sharing one with line voltage always is. The reason turns on what a bridle ring is and what the separation rules actually cover.
What the separation rules cover
Section 760.136(A) states that cables and conductors of power limited fire alarm circuits shall not be placed in any cable, cable tray, compartment, outlet box, device box, raceway, or similar fitting with conductors of electric light, power, Class 1, non power limited fire alarm, or medium power network broadband communications circuits. Section 760.136(B) then requires a minimum of 2 inches of separation from those conductors where they are not in a raceway, unless one of the listed conditions is met, such as the power conductors being in a raceway or in metal sheathed cable.
An open bridle ring is a support device, not a raceway, enclosure, or fitting in the sense used above. That is why the code language does not squarely prohibit two low voltage cable types occupying one ring. What it does prohibit is running any line voltage NM or MC cable through that same ring alongside power limited fire alarm cable without the 2 inch separation.
Mixing fire alarm with other power limited circuits
Section 760.139 permits Class 2, Class 3, and communications circuits to occupy the same cable tray, raceway, or enclosure as power limited fire alarm circuits under the conditions given there, generally requiring the other circuits to be installed with the same requirements as the fire alarm circuits. In practice, a data cable and a fire alarm cable in the same ring is not the violation that gets written up.
Where jobs actually fail
Three things get cited more often than the ring itself. First, 300.11(A) requires the raceway or cable support to be securely fastened in place, and 300.11(B) prohibits using ceiling support wires as a support for the cable. A bridle ring hung from a grid wire is a violation regardless of what is in it. Second, 760.124 requires the fire alarm cable and its supports to be identified, which is why red jacket is the convention and why some jurisdictions require a red support or red tag. Third, NFPA 72 and many specifications require fire alarm cable to be independently supported so a data crew cannot disturb the life safety system during a future move.
Field takeaway
Keep line voltage 2 inches away or in metal, do not hang the ring from ceiling grid wires, and expect many inspectors and specifications to want the fire alarm on its own supports even where the NEC would allow sharing. Ask the AHJ before you install a thousand rings.