Aloha!
Always nice to see some more Chocobo Theorycrafting!
Unfortunately there's a problem with Chocobos and Aggro Control.
Basically, Chocobos can't Control Aggro.
Your Chicken also has a very low HP pool (Base HP is 1943 at level 50) making his ability to withstand spike damage and passive HP regeneration both fairly low. Going for a "full defender" spec will therefore result in a Bird that has low damage output, low HP, poor defenses, and can't reliably hold hate.
The Chocobo will only focus on one enemy at a time, so he cannot tank + build hate on multiple mobs simultaneously. Whilst the Rank 10 attacks from Defender and Attacker Stance would technically allow him to hit more than one enemy, they are both Melee Cones rather than PBAoEs and cannot be triggered manually. As such, they will very likely be "wasted" by the poor Chocobo battle AI not lining itself up properly.
In order to get a more hardy bird that generates maximum levels of hate, you really want the first attack from the Defender tree (Choco Drop - the only one with an enmity modifier until Rank 10 - you want the Chocobo to SPAM this) combined with the maximum amount of passives granting +Damage and +HP buffs possible.
Realistically, taking Rank 3 Defender/Rank 9 Attacker/Rank 2 Healer would be the "best" option here for a Tanking Chocobo - large amounts of HP and STR, buffed Autoattack speed and access to +Regen.
Going for a "hybrid" rather than "pure" build is generally going to be the most efficient use of SP, because careful consideration has to be made in terms of benefit granted per point of SP spent. The Rank 10 Defender/Attacker abilities are both a very poor investment in this regard - they deal underwhelming damage and the AI is very unlikely to line itself up properly in order to hit multiple mobs. Unfortunately the Rank 10 Healer ability is very likely to be underwhelming as well due to its rather low healing potency combined with the aforementioned poor AI.WAR - STR 10%, HP 20%, ACC 5%, Crit 15%, Damage Reduction -15% (w/ AOE Enmity Attack)
PLD - STR 15%, HP 25%, ACC 5%, Crit 15%, Damage Reduction -15% (w/ Singular Attacks)
PLD2 - STR 10%, HP 25%, ACC 5%, Crit 15%, Damage Reduction -15%, MND 10% (w/ Cure)
WHM/SCH - MND 25%, HP 10%, ACC 5%, Crit 15% (w/ AOE Cure)
BLM/BRD - STR 15%, HP 5%, ACC 5%, Crit 15%, Attack Speed 10% (w/ AOE Attack)
MNK/DRG - STR 20%, HP 15%, ACC 5%, Crit 15%, Attack Speed 10% (w/ Singular Attacks)
SMN - STR 10%, HP 20%, ACC 10%, Crit 30%, MND 10% (w/ Highest Crit + Status Effects + Cure)
As mentioned in another thread, there are only four Hybrid builds I would suggest, and three that I would actually recommend:
Dedicated Tank role: 3 Defender, 9 Attacker, 2 Healer.
(+20% STR, +15% HP, +5% Acc, +15% Crit, +10% Attack Rate, +10% MND. 4 Weaponskills and Regen)
As mentioned earlier. To Tank, run in "Defender Stance" to keep it spamming 'Choco Drop' for maximum +enmity generation, and flick in-and-out of "Free Stance" occasionally in order to make the Chocobo cast Choco Regen on itself. Note that even with this setup, the Chocobo will struggle to hold hate over you, so overall this is a rather weak Companion build. NOT RECOMMENDED.
Dedicated DPS role:4 Defender, 9 Attacker.
(+20% STR, +15% HP, +5% Acc, +15% Crit, +10% Attack Rate. 5 Weaponskills)
This is identical to your "Monk/DRG" suggestion above. Has maximum number of available Weaponskills, a large amount of +Str and +HP from passives and an Autoattack speed buff. Always run in "Free Stance". This build should only be used if YOU are prepared to handle all healing. Note that the Rank 7 Attacker ability has a 'Stun' effect and can be very annoying in certain battles due to Diminishing returns. RECOMMENDED (gives highest possible DPS, but assumes that you are always going to play as a healer)
Hybrid DPS/Healer role: 9 Attacker, 4 Healer.
(+15% STR, +10% HP, +5% Acc, +15% Crit, +10% Attack Rate, +10% MND. 3 Weaponskills, Regen and Cure)
Very self-sufficient. You can put the Chocobo into "Attacker Stance" for maximum DPS, and "Healer Stance" for maximum healing potential, but leaving it in "Free Stance" will give good performance at both. Same warning as above regarding the Stun and Diminishing returns. RECOMMENDED (Slightly higher DPS, but assumes that you are NEVER going to want your Chocobo to tank and are prepared to put up with Stun DR).
All-Rounder Build: 7 Healer, 3 Defender, 6 Attacker.
(+10% STR, +20% HP, +10% Acc, +30% Crit, +10% MND. 3 Weaponskills, Regen, Cure and +30% Healing Buff)
Extremely versatile and self-sufficient. Put into "Defender Stance" when you want it to try to Tank, or "Healer Stance" whenever you want it to only concentrate on Healing... otherwise leave it in "Free Stance". The Rank 7 Healer buff is very high uptime, so is definitely worth taking if possible. This setup has good DPS whilst avoiding the aforementioned Rank 7 Attacker "Stun" Weaponskill (and the Rank 4 Defender "Slow", which can also trigger Diminishing Returns). RECOMMENDED (Decent DPS and useful for basically everyone, with no danger of a badly-timed Stun inflicting DR).
I originally had a few ideas for a "Maximum Support" build for Tanks or Melee DD characters... but about the only thing you could take over the All-Rounder Build to help with this would be the Rank 4 Defender tree Weaponskill. Taking that would negatively impact the already-poor tanking capability of the Chocobo, and the trade-off (dropping the +15% Crit from the Attacker tree) isn't really worth it either since Crit buffs affect both attacks and heals.
It should probably also be mentioned that "Healer Stance" is not always the best option for healing, especially if you are a Melee character. Putting your Chocobo into Healer Stance will raise the threshold at which your Chocobo will start spamming Heals on you (~80% HP), but the bird will stop autoattacking and follow your character - making it get hit by any enemy AoEs and Cones aimed at your character, which results in your Chocobo having to regularly heal both you and itself. Leaving your Chocobo in "Free stance" will lower the threshold at which it starts casting heals on you (~50% HP) but keep it autoattacking - making it far easier to position it out of the range of enemy Cones/AoEs, which results in it being able to focus only on healing you.