Your method certainly makes the rotation more consistent, but refreshing DoTs right before RS ends (or any other helpful buffs from a party member) means that for the next 30 seconds your DoTs are still doing 20% more damage (or have a higher crit rate, etc. in the case of other party buffs). The way DoTs work is it takes a snapshot of your (and the target's) buffs/debuffs at the time the DoT is applied and continue ticking at that potency even after those buffs/debuffs fall off of you. This means that your method has your DoTs ticking with the RS buff for 30 seconds (the length of the DoT), whereas reapplying your DoTs right before RS falls off means that those DoTs are now going to have the RS buff tick for ~50 seconds (reapply at the start of RS, ticks for 20 seconds, reapply at the end of RS, ticks for another 30 seconds).
An extra ~20 seconds of buffed DoTs means that you wind up doing an extra ~120potency worth of damage. It's not huge, but more damage is more damage.
Math:
Bard DoTs are 55 + 45 = 100potency per tick.
RS is +20% damage, so under RS you get an extra 20 potency per tick.
Assuming an extra 18 seconds of buffed DoTs because DoTs tick every 3 seconds, so 6 extra DoT ticks.
20potency * 6ticks = 120 extra potency.
With all that said, unless you're raiding, the difference is obviously minimal. If you do this, like you said, you will lose the synchronization of DoTs with your SS and song timers.
Unrelated question...
Is there a reason why the suggested opener seems to be DoTs > SS > then IJ? I've never really seen a BRD discussion anywhere, and I always do SS > DoTs.
Specifically (if you want to poke holes in my opener), SS > RS > BL > SB > Song > EA > CB, then refreshing DoTs before RS falls.
