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.
P6 chip. Triple the speed of the Pentium.
Tualatin: PIII-S @ 1628MHz | QDI Advance 12T | 2GB DDR-310 | 6800GT | X-Fi | 500GB HDD | 3DMark01: 14,059
Dothan: PM @ 2.9GHz | MSI Speedster FA4 | 2GB DDR2-580 | GTX 750Ti | X-Fi | 500GB SSD | 3DMark01: 43,190