As a fighter, you do not have a spellcasting class.
You quote the relevant text from Golgari Agent:
Golgari Guild Spells
Prerequisite: Spellcasting or Pact Magic class feature
For you, the spells on the Golgari Guild Spells table are added to the spell list of your spellcasting class. (If you are a multiclass character with multiple spell lists, these spells are added to all of them.)
Without levels in a class with the Spellcasting feature, you do not have a spellcasting class, so the Golgari Guild spells are not added to any spellcasting class's spell list for you. The key distinction here is that being able to cast a spell from a feat is not the same as having the Spellcasting class feature (see my answer here for more details). Therefore, the Guild Spells feature does not change which spells you may select with Magic Initiate.
Going strictly by the book, to select entangle with Magic Initiate, you must do one of the following things:
- Select Druid when you take Magic Initiate.
- Take at least one level in a spellcasting class so that Golgari Agent adds entangle to that class's spell list, then select that class when taking Magic Initiate.
- Take the Eldritch Knight martial archetype at 3rd level, which adds entangle to the Wizard Spell List, then select Magic Initiate at 4th level, selecting the Wizard class for the feat.
There is no reason not to allow it anyway.
Despite the rules as written not working out, ask your DM. The guidance given in the Dungeon Master's Guide for small changes to spell lists is essentially "it probably isn't a big deal, be careful with warlocks":
Modifying a class’s spell list usually has little effect on a character’s power but can change the flavor of a class significantly. In your world, paladins might not swear their oaths to ideals, but instead swear fealty to powerful sorcerers. To capture this story concept, you could build a new paladin spell list with spells meant to protect their masters, drawn from the sorcerer or wizard lists. Suddenly, the paladin feels like a different class.
Be cautious when changing the warlock spell list. Since warlocks regain their spell slots after a short rest, they have the potential to use certain spells more times in a day than other classes do.
Since we're talking about the spells chosen with Magic Initiate, I cannot foresee any balance issues with letting you pick entangle for your 1st level spell, and I would be inclined to allow it. And as far changing flavor, in your case, this would be a step toward the appropriate flavor for your character, since you would be taking a spell used by members of your guild. This is a case where rules as written, it doesn't work, but there really is no practical reason not to allow it.