spring talon
As can be seen in the video, I instantly regained my Relic on use, due to the Medic Prime Perk, but still lost my 20% Damage Bonus, and incurred the -20% Damage Reduction penalty. This doesn't reset until after the 30 second Relic recharge, despite a Relic already being present. The description of Cost of Betrayal states that the bonus or penalty being applied is dependent only on whether or not you have a Relic available, not on the 30 second Relic recharge.
I tested this with, and without, the Burden of the Departed Ring, as well as with other Relics, and the results were always the same.
You can also see how the Relic recharge doesn't store a charge, like it normally does, when your Relic capacity is full, but instantly rolls over, instead, as if it recharged a missing Relic. This also occurs when wearing just the Burden of the Departed.