Banner of command (promote) and baron buff. Which minion should be promoted?

So when looking through the buffs, I came to the conclusion that in a setup where there is no champions involved (except for one that has baron buff, so minions actually have the bonus), just minions, the following would apply (I will get onto champions later):

which minion would make destruction of towers faster

Except for caster minions, baron buff would be more suitable as:

Melee minions

• (Gain +50% movement speed when within 800 units of enemy minions or turrets)
• (Increased size)
• (+75 attack range)
• +75% damage reduction versus champions and minions
• +30% damage reduction versus turrets (similar to cannon minions)

Cannon minions (beyond turret range)

• +600 attack range
• +50 attack damage, but attack speed is halved
• Attacks are now Area of Effect (200 range) and deal double damage to turrets

Quite important to note: Cannon minions actually attack turrets from out of range, so they will shoot infinitely as long as other minions do not kill them.

This does not apply to Caster Minions as they receive similar offensive buffs to the ones of Banner Of Command, but the defensive stats from Banner of Command will make destroying turrets easier as the minions survive longer:

• • 400 Health.
• • 40 Armor and Magic Resist

and which would be more suitable for clearing minions

Cannon minions (beyond turret range)

• Attacks are now Area of Effect (200 range) (and deal double damage to turrets)

The AoE will clear minions faster, considering they will stand in standard 3-1-3 or 3-0-3 formation (melee-cannon-caster)

Meelee minions (in a 1v1-scenario) empowered with Banner Of Command should overcome minions that have Hand Of Baron as they do not receive major offensive boosts from Hand of Baron in contrast to Banner Of Command granting:

• +50 Attack Damage
• +90% bonus Attack Speed

The defensive boost are comparable. While Hand of Baron gives +75% damage reduction versus minions, Banner Of Command gives

• +600 Health
• +40 Armor and Magic Resist

Caster minions (again, in a 1v1-scenario) should overcome with Banner of Command as the offensive statistics of both buffs are comparable, but Banner of Command gives bonus health and resistances.

DISCLAIMER: In my eyes, this is rather hard to theorycraft, as the circumstances under which you apply the buffs decide the outcome. Questions as:

Do the minions come in waves or are they single-handedly going onto towers?

Are their champions with waveclear or which turrets (Outer or Inner Turrets) are attacked?

Context: While baron gives resistance to slow effects, 75% damage reduction versus area of effect, damage over time, and persistent effects, Banner of Command gives a permanent Black Shield which grants total immunity to magic damage and Tower Shield (reduced turret damage). Turrets prioritize minions that have been promoted over other minions. So it is important to notice that there is a lot of factors to be considered.

Personal opinion should not be included, but considering that Hand Of Baron gives the buff to all nearby minions and not just one unit, I would personally prefer it over Banner of Command. In a scenario where a Baron-empowered wave faces a wave with a single Banner of Command-empowered minion (choose whichever), I am certain the wave with baron will push.

I suggest you refine your question and/or play a custom game and try out your idea yourself. ;)