First off, Mr. Happy's 2.1 guide is outdated and not the best source to go by (heresy, I know, but bear with me here).
Second, ditch macros. Regarding the way they work, they're a DPS loss and encourage senseless button mashing. You want to be fluid and concise with your inputs.
With that out of the way, your opener should look like this (which I got from a pretty damn good BRD):
DoT1 + Bloodletter > DoT2 + Hawk's Eye* > Straight Shot + Raging Strikes + Blood for Blood > Heavy Shot + Barrage** + Internal Release > Heavy/Straighter Shot + Flaming Arrow > Heavy/Straighter Shot filler until 2-4s remaining on your first DoT. Reapply your DoTs before they fall off and go back to your priority system.
*If your skill speed isn't too high, or there isn't a move that needs to be silenced, you can weave in Blunt Arrow right after Hawk's Eye.
**If you are using high-quality X-Potions of Dexterity, you might want to use it before Barrage. The potion greatly boosts your damage output, and Barrage grants us our biggest damage spike with three auto-attacks in a single auto-attack.
Here's a list of priorities BRDs should watch for:
1st) Use Bloodletter in-between GCDs whenever it is off CD. Should the recast timer refresh during your opener, use it and keep on going. Your DoTs will still be reapplied with all buffs still up regardless.
2nd) Reapply DoTs before they fall off. Letting DoTs fall off is not only a DPS loss, but you lose out on potential Bloodletter resets, which helps contribute to our spikes in DPS.
3rd) Reapply Straight Shot's buff before it falls off. Reapplying DoTs with the buff up improves your chances at getting more Bloodletter resets.
4th) Does Straighter Shot proc after using a Heavy Shot? USE IT. Guaranteed critical damage is always welcomed, and using Straighter Shot refreshes the Straight Shot buff.
5th) Make sure to face the target whenever possible. Auto-attacks contribute to our DPS and not facing the boss means a chance to lose out on that damage.
Before anyone bashes me for not suggesting the traditional Straight Shot opener, opening with either Windbite or Venomous Bite and following up with Bloodletter right after, in-between GCDs, exposes you to potential Bloodletter resets right from the get-go. That means you already have the chance for a 150 attack potency move to reset its cool-down and be ready to weave in again. On top of that, you reapply your DoTs with all of your buffs up, so you don't have to lose DPS to clip them before the buffs fall off. With the Straight Shot opener, you have to get a Straighter Shot proc within 3-4 GCDs or you have to reapply it before clipping your DoTs, or reapply your DoTs without the buff up which defeats the purpose of the opener.
Now for cool-down rotations, which are used throughout long fights like trials and raids. I credit our resident BRD Ooshima for this rotation of cool-down groupings, which I copied and saved in a text file months ago while lurking.
LEGEND
RS = Raging Strikes
HE = Hawk's Eye
B = Barrage
BfB = Blood for Blood
IR = Internal Release
FA = Flaming Arrow
XD = X-Potion of Dexterity
CD Groups
G1: IR + FA
G2: BfB + HE + B
G3: RS
Weave Rotation
Start: The opener I provided
60s: G1
80-90s: G2
120s: G3 + G1
180s: G1 + G2
240s: G3 + G1
270s: G2 (+XD between HE and B)
300s: G1
360s: Go back to Start (use XD when available)
With this rotation of cool-down groups, your overall DPS should see some definite improvement. Remember to keep your DoTs up, use Straighter Shot and Bloodletter whenever they proc, face your targets. Oh, one more thing: be mindful of how you use Repelling Shot and always sing Foe Requiem right when the tank begins the pull and turn it off after that pull is cleared. You don't want to have "loldrg" (or is it "lolnin" now?) moments with it. As for Requiem, it is the only song that doesn't nerf our damage while its up and boosts all caster damage. Even if you don't have caster DPS (BLM, SMN) in your party, sing it either way in case you get a healer who also helps with DPS when they don't need to heal. Just remember to turn if off after each pull or otherwise you'll pull the mobs instead. A lesson I learned the hard way. Sing the other two songs when needed.
I hope this helps you out, OP!