I don't think there will be any problems burning BDs with that system.
The slowest system I burned a BD-R on was a S939 system with a dual core Opteron at 3GHz. I was burning a BD movie image stored on my file server. The system was pulling the file over GbE, which probably increased CPU load a bit. Burning at the drive's max speed (12x or 54MB/s, if the 1x = 4.5MB/s stated above is correct) CPU load was in the 10-15% range if I remember correctly.
Based on that, and taking into account the difference in CPU performance (I'm also assuming that your source file is stored on a local HDD), I think a dual P3 1GHz would be just fine burning a BD, even at the drive's maximum speed.
Standard Def Rigs
Super P3: PIII-S @ 1.63 GHz/FSB155 | 2GB DDR-310 | 6800GT AGP | 500GB 7200 RPM
Super G4: 2x PowerPC 7455 @ 1.5 GHz | 2GB DDR-333 | 7800GS AGP | 300GB 10k RPM
Super G5: 4x PowerPC 970 @ 2.5 GHz | 16GB DDR2-533 | x1950XT PCIe | 512GB SSD