Allow me to explain why I prefer opening with DoTs over Straight Shot.
1) Starting with Straight Shot means missing out on a potential Bloodletter reset, a 150 potency attack.
2) If you start with Straight Shot you need a Straighter Shot proc in about 3-4 GCDs (10% chance isn't exactly in your favour). If you don't get that proc, your options are as follows: a) reapply Straight Shot early, wasting potency and TP over a Heavy Shot (70TP for a 140 potency move vs. 60TP for a 150 potency move); b) clip your DoTs (again, a waste of potency); or c) reapply DoTs without Straight Shot which defeats the purpose of the Straight Shot opener completely.
3) With the opener I use, Windbite and Venomous Bite are reapplied with all of the buffs up without having to clip them. On average, I reapply my first DoT whenever it has 2-4s remaining, and by the time my second DoT is reapplied my first buff used, Hawk's Eye, still has 4-6s of duration left before it falls off. You don't need to have your first set of DoTs buffed, and besides...
4) With the traditional Straight Shot opener, to stack your buffs and get the biggest increase you can on your skills, you would have to hold them all long enough that you'd end up losing DPS. From another perspective, if you were weaving your cool-downs as soon as possible, you would only have some buffs up for your first set of DoTs and some buffs up for your second set, which isn't as good as having them all up for your second set of DoTs and none at all for the first.
As for Quelling Strikes, I omitted it due to its usage being on a case-by-case basis. If you have a tank who can hold aggro against top-notch DPS, there is no need for it other than to buy breathing room. If a powerful add needs to be grabbed by the tank, Quelling Strikes can help let the tank grab it safely (i.e. Clockwork Dreadnoughts in T8). If you need Quelling Strikes for anything outside of adds or speedrun pulls, that says more about the tank's ability to hold aggro instead of the lack of using Quelling Strikes. This is coming from someone who runs both BRD and WAR on a frequent basis.
This is the gear I use outside of SCoB, by the way: http://ffxiv.ariyala.com/O5A5
No problem. If missing Straighter Shot procs and not reapplying DoTs before they fall off are your only issues, that should be quick to remedy. After using Heavy Shot, wait a bit before using it again to see if you get the proc. With a 2.4s-2.5s GCD (depending on your skill speed), that should give you plenty of time to wait for a proc. For DoT reapplication, the best time to refresh them is around 2-4s on the first DoT. As DoT damage is based on server ticks and not the duration of the DoT itself, it's best to reapply them around that time lest you miss a tick soon after a DoT falls off.
Good on you for double-buffing, by the way! It gets tougher to do with higher skill speed and animation locks on certain skills, but damn is it good. Maybe Square will reduce the animation locks for a lot of skills like they did with the DRGs Jump abilities this patch one day? Anyway, keep working on your BRD and continue to strive for improvement!