Let us say you did 300 dps over the course of a 5 minute encounter. DPS is measured in seconds, so that's 60*5 = 300 seconds. 300*300 = 90000 damage.
Now, singing Requiem takes up 3 seconds. That's 900 damage lost if we assume that it's a raw conversion. 89,100 damage done.
Now, let's assume (Even though the BLM will be doing more than you) that the BLM is only doing 300 DPS as well, over the same 5 minutes. Again, 90,000 damage.
Let's say it lasts for 30 seconds with 10% extra damage. the BLM will be doing 300 DPS for 4 minutes 30 seconds, and 330 DPS for 30 seconds.
300*270 + 330*30 = 9,900 DPS
In just that 30 seconds alone, the BLM has already made back the DPS you lost. And these are with numbers that extremely favor the BRD's DPS over the BLM's. Your Foe's Requiem will last more than 30 Seconds, and this doesn't account for using BV at the start.