The beta lists "L1.91 for Update CPU code to support AM3+ CPU." Where did you get "BD"? Ref - http://www.asrock.com/mb/beta.asp?Model=770%20Extreme3
'My thoughts' are normally I only recommend 970~990FX chipsets for the FX CPUs so folks can obtain the optimal performance. Your MOBO can support up to a 140W CPU, so if the beta BIOS includes support for the FX-8120 then it should work.
Problem, the ASRock 770 Extreme3 isn't a stellar MOBO to start out with and so besides it 'running' the FX-8120 @ stock my deeper concerns are OC'ing a very OC'able CPU...
General reasons for updating a BIOS, this falls under (#1) reason:
In short, if you have no problems then my best advice is NOT to flash.
Examples 'when to flash':
1. CPU requires newer BIOS version
2. Non-OS Instability
3. Peripherals e.g. 2TB HDD requires new BIOS.
4. Memory compatibility ; where current RAM is causing errors or stability