The beginning of PotA states the following:
"Princes of the Apocalypse presents an epic campaign against the forces of Elemental Evil that can occupy you and your players for many months. Characters who are at least 3rd level can dive right into the main adventure. A party of 1st- or 2nd-level characters should start with the mini-adventure in chapter 6."
Clearly, if you are starting at 1st level then the adventure is intended to start with chapter 6 so there shouldn't be any problem there.
As for changes, an AL DM has a great deal of latitude but the goal is to stay consistent with the adventure as laid out. Cosmetic changes and tweaks to make it fun for the players are generally considered ok. Rewriting the adventure to suit an individual DMs perspective or preference is not.
From the ALDMG:
"
Being the Dungeon Master
You have the most important role—facilitating the enjoyment of the game for the players. You provide the narrative and bring the game to life.
You’re Empowered. Make decisions about how the group interacts with the adventure;
adjust or improvise as needed. It’s more important that you and the players are having fun than you stick to letter of the adventure.
Everyone Should Shine. Don’t allow any one player to dominate the table. Everyone should get the opportunity to shine. Feel free to adjust the encounters and adventure to make this happen.
Keep the Adventure Moving. This is mostly for convention play where you’re participating in a timed slot. When the game gets bogged down, provide hints and clues to your players facing puzzles or engaging in combat and roleplay interactions to help move them along."
So, the AL DM is allowed to tweak or adjust as they see fit to keep the adventure fun and interesting. However, they shouldn't make changes just because they feel like it.
For example, why swap a bear for a zombie bear in "Into the Wilderness"?
Spoiler: PotA spolier
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The text in the adventure says the bear is supposed to flee when reaching 1/2 hit points. This is because a bear has multiattack and a single round of attacks from a bear is quite capable of killing a first level character. A crit from a bear can easily cause perma death to a first level character.
If you want to call the bear a zombie bear just for role play reasons without giving it any of the features of a zombie, having it still flee at 50% hit points and attack whoever is closest whether it is a PC or a bandit then there is no real issue with just changing up the "fluff".
However, if you are actually swapping a bear for a zombie bear it does several things ... 1) A zombie isn't about to flee, it doesn't care about damage - this increases the likelihood of a PC dying or possible TPK for first level characters depending on the classes and how many there are ... 2) A zombie has the feature Undead Fortitude. This means that even if you get it to zero hit points it makes a con save to see if it stays up at 1HP. Zombies can be irritating to kill off even at higher levels as they make these saves and keep on fighting. Keeping these two features in mind, swapping a zombie bear for a regular bear is almost guaranteeing one or more characters will die unless you are the type of DM who carefully manages the dice and the targets chosen by NPCs to avoid this outcome. 3) If the zombie bear was given to the bandits by the necromancer then it is unlikely to attack the bandits. If it comes into the fight as an ally of the bandits then the situation gets even more lopsided.
I'm also not sure I see the point to the other changes.
New Management is a side quest and the motivations of the NPCs involved are pretty well outlined.
Spoiler: PotA Spoiler
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Stealing money from the Zhentarim to pay protection money to a cult on behalf of the village just seems out of character unless you change the villain of the module to be someone else. However, then all of the actions that require an "inside" presence to make them work don't really fit either.
In addition, the point of New Management is a frame job by the real culprit. They don't really want multiple possible options for the person who steals the gold. They want the players and everyone else to leap to the one conclusion they are staging with the various bits of evidence. You could re-write he entire thing to have a different point but I am not sure why it would be needed nor how it would increase the enjoyment of the encounter as written.
However, most of the tweaks you are considering are cosmetic and as a result would not be banned for AL play. The AL DM shouldn't be changing up the adventure but modifying bits of the descriptions to make the story hang together in a particular way without modifying either the encounters or magic items unlocked should be fine.
Also, keep in mind that AL rewards will be Advancement Check Points (1/hr of play), Treasure Check Points (1/hour of play in tier 1), Downtime (2.5 days/hour of play) and renown (0.25/hour of play). The adventures do not award any extra gold and magic items found will need to be purchased using TCP with the exception of some important items that are considered Storyline awards while the character is playing that particular hard cover.
Anyway, since PotA is supposed to start in chapter 6 for first level characters, there is no problem starting with that introductory adventure in AL.
