Technically, the rules-as-written used to indicate that such a dragon would have a 0d6 breath weapon of the appropriate energy type. This was important because certain abilities can add damage to attacks using a certain energy type or activate when making a certain kind of saving throw successfully. Because the ability specifically calls out 'racial HD' as opposed to just 'HD' the normal HD/class levels equivalency rule most certainly does not apply, so your GM is soundly wrong on that point.
However, James Jacobs, the Paizo creative director, indicated that half-dragons without racial HD should not be given a breath weapon at all here, where he also cited his hatred of half-dragon humanoid characters and fervent desire that people would stop making them as the goal in this decision.
As creative director, James Jacobs is pretty much in charge of the rules, and the pathfinder errata model supports random posts on the Paizo boards made by Paizo staff as a form of authoritative text with errata power, until replaced by a more formalized FAQ post, official errata, or newer printing. Usually if one of these things is planned or considered, the staff member posting the forum post will indicate that the change will be noted in a future source. No such indication was made with the post, so it is presumably intended to stand as authoritative errata on its own.
There are a lot of sources which are almost universally regarded as RAW by among the Pathfinder community, such as the Core Rulebook. There are a lot of sources that are not regarded as RAW equally universally, such as random posts on the Paizo boards by non-Paizo staff members and answer text found on this site. Non-FAQ errata posts are an area of contention, however, as a few groups that play RAW don't consider any material not published in print RAW and many (but still a small minority) groups only consider 'official' errata (these) to be valid.
The point here is that both a 0d6 breath weapon and no breath weapon at all are possible RAW interpretations, depending on your scope for RAW, with the latter being the only correct choice in the pathfinder errata paradigm.
The Green Dragon's breath is described as:
Poison Breath (Recharge 5-6). The dragon exhales poisonous
gas in a 90-foot cone. Each creature in that area must make
a DC 22 Constitution saving throw, taking 77 (22d6) poison
damage on a failed save, or half as much damage on a
successful one.
It is "poisonous gas" the Dwarf gets advantage on the save.
Poison. Venomous stings and the toxic gas of a green
dragon’s breath deal poison damage.
It is "poison damage" the Dwarf takes half damage (effectively 1/4 if saved 1/2 if not).
With respect to your comments on protection from poison, I think you are splitting hairs: it is poison so advantage is given on the save; the fact that the effect is to do poison damage rather than impose the poisoned condition is irrelevant to the save.
That said, the DM is the sole authority on rulings so, check with your DM.
Best Answer
Let's start with your incorrect premise: an Area of Effect does not go around corners (unless specified by the effect). There must be a clear path to the point of origin for the effect, and then a clear path from the point of origin to the affected creature. This might seem like going around corners but is actually two straight lines. [See Area of Effect and Cover, DMG p251.]
In the case of dragon's breath, the point of origin is the dragon, with the Area of Effect being a ray or cone. Thus, there must be a clear line from the dragon to the target.
Arrow slits provide 3/4 cover [DMG p251]. Dragon's breath would come through the slits. If there are places on the wall where you could stand away from the arrow slits, those spots would have total cover (see below diagram for a cone AoE).
Thus, if you are standing at the arrow slit casting spells or shooting at the dragon when it uses its breath weapon, you would need to make the required save.
By RAW, 3/4 cover gives +5 to AC and Dex saving throws [PHB p196]; however, a DM may house rule advantage instead of +5 on the save if a square with total cover is next to you.