First, about the feature's wording.

The wording of the Symbiotic Entity feature that you quoted doesn't completely match either the Unearthed Arcana version of the Circle of Spores or the later published Guildmaster's Guide to Ravnica version, although it is very similar to both.

Here's the version from Unearthed Arcana: Three Subclasses, where it first appeared:

At 2nd level, you gain the ability to channel magic into the spores that infuse you.

When you use your Wild Shape feature, you can awaken those spores, rather than transforming. When you do so, you gain 3 temporary hit points per level you have in this class, the damage of your Halo of Spores feature doubles, and your melee weapon attacks deal an extra 1d6 poison damage to any target they hit. These benefits last for 10 minutes or until you use your Wild Shape again.

And here's the Guildmaster's Guide to Ravnica version (this is fair use):

At 2nd level, you gain the ability to channel magic into your spores. As an action, you can expend a use of your Wild Shape feature to awaken those spores, rather than transforming into a beast form, and you gain 4 temporary hit points for each level you have in this class. While this feature is active, you gain the following benefits:

• When you deal your Halo of Spores damage, roll the damage die a second time and add it to the total.

• Your melee weapon attacks deal an extra ld6 poison damage to any target they hit.

These benefits last for 10 minutes, until you lose all these temporary hit points, or until you use your Wild Shape again.

So you might want to check the source you got your quote from to make sure you have accurate versions of this and other subclass features you use. Your quote looks like it was made by someone updating the UA version with the changes in the Ravnica version without quite matching the wording, and that could be problematic if changes were overlooked anywhere else in the subclass.

Regardless, the same key wording ("rather than transforming") is used in your quote, in the UA version and in the Ravnica version, so this answer is effectively universal. You might want to check the source you got your quote from, however, to make sure you have accurate versions of all your subclass features.

Yes, you ignore the downsides.

If you use your Wild Shape feature to do something else "rather than transforming" then you are literally not transformed after that, opting for another benefit in its place, in this case awakening your spores. Thus any restrictions you would normally be subjected to "while you are transformed" would literally not apply.

Note the following wording in the druid's Wild Shape feature (emphasis mine):

While you are transformed, the following rules apply:

This precedes the list of benefits and restrictions you are subjected to when you are in a beast form, including the restriction of being unable to cast spells. These would not apply unless you're actually transformed.

If they were intended to apply just because you used the Wild Shape feature even if it didn't involve transforming, then it would say "while using this feature" or "while benefiting from this feature" rather than "while you are transformed." (And the designers would have been unreasonable to publish the Circle of Spores with the wording used if there were such an ambiguity.)

Effectively, this allows the Circle of Spores feature to utilize your Wild Shape uses as an expendable resource without creating a new resource just for that purpose (the Ravnica version of the feature makes this explicit) and without stacking the benefits of awakening your spores on top of the benefits of being in beast form.

There are few reasons the poison damage probably does not apply to the splash damage target

1. Symbiotic Entity requires you to actually hit your target.

   The Symbiotic Entity feature states:

   Your melee weapon attacks deal an extra 1d6 poison damage to any target they hit...

   The books use the term "hit" not as its standard English meaning but to describe when an attack roll exceeds or equals the AC of a target creature. The secondary effect of green-flame blade does not involve making an attack, and so you have not actually hit this second target.

2. The secondary damage of green-flame blade likely is not part of the actual melee weapon attack:

   The Symbiotic Entity feature states:

   Your melee weapon attacks deal an extra 1d6 poison damage to any target they hit...

   One way we could hope to know an answer here is by looking at the section on "Critical Hits" which states:

   When you score a critical hit, you get to roll extra dice for the attack’s damage against the target. Roll all of the attack’s damage dice twice and add them together.

   We know that critical hits only modify the attack's damage so if a critical hit doubled green-flame blade's secondary damage, that damage would have to be considered part of attack's damage and thus part of the melee weapon attack required by Symbiotic Entity. Unfortunately this is not very clear...

   One thing we do know is that damage gated by a saving throw does not double when you score a critical hit. This is supported by this Q/A ("Is Ice Knife's explosion damage affected by rolling a critical?") and also this Q/A ("How does extra damage work for critical hits?"). This is also supported by lead game designer, Jeremy Crawford, though his idea are not official rulings:

   Q. DMG Poisons: If you crit with a poison coated weapon, do you double poison dice because crit, or not because saving throw?
   A. The intent is no. The saving throw, not the attack, determines whether the poison takes effect after a hit.

   However, with green-flame blade there is no save that is gating the additional damage, so perhaps critical hits would double its damage like they do with sneak attack. Crawford has also had input on this specific question stating:

   Q. If you critically hit with the attack part of Green Flame Blade, and are greater than level 5, do you roll extra dmg against the 2nd target?
   A. The splash damage of green-flame blade isn't affected by the attack critting. Think of the attack as process X & the splash damage as Y.

   Thus we at least know the intent is that green-flame blade will not double it's additional die when critting meaning it does not count as part of the attack's damage. Because of this we can conclude that it is not part of the melee weapon attack and thus cannot cause the additional 1d6 damage from Symbiotic Entity.

3. Symbiotic Entity only effect a target of a melee weapon attack not all of its targets, plural.

   The Symbiotic Entity feature states:

   Your melee weapon attacks deal an extra 1d6 poison damage to any target they hit...

   Because this feature only applies to a singular target which we hit, it cannot apply to more than one person affected by our attack. This idea also would allow us to rule that critically hitting with green-flame blade does not double the additional damage against the second target because critical hits have the same wording, using the singular form of "target".

A note: This does of course require that we assume the phrase "any target" is, in fact, singular. As user @thedarkwanderer has pointed out this phrase could be read as already being plural in which case my argument for point 3 would not apply. There's ambiguity in the meaning as the word "any" in English - It can be used as the plural over the alternative options of "each/every".

A small note on the exceptional state of green-flame blade:
This spell is rather exceptional; you make only one attack but end up affecting multiple creatures. The Sage Advice Compendium has said the following regarding the Ranger's Whirlwind Attack and Volley features:

Whirlwind Attack is unusual, in that it’s a single attack with multiple attack rolls...

These two features and green-flame blade are the only cases I know of where a single attack affects multiple creatures and so they probably have some odd interactions such as the one asked about in this Q/A ("How does the Sharpshooter feat interact with the Hunter ranger's Volley feature?")