Since you already know the year, and are using the Forgotten Realms Calendar Tool to pick the day based on the lunar cycle, I intend to restrict this answer to solely determining the month. Also, I'm not going to bother using the Forgotten Realms versions of month names because doing so would make it harder to follow my logic. I'm leaving the conversion up to the reader.
The best indication for what month it is when Hoard of the Dragon Queen begins is the information provided regarding Greenest's Mill.
The south-western region of the Realms (which would include Baldur's Gate, Candlekeep, Beregost, Elturel, and most importantly Greenest) has a temperate climate. The air is very humid in this region, and it sees a great deal of precipitation. Additionally, volcanic activity is fairly common. In fact, Candlekeep itself is built on top of a volcanic crag. This information was obtained from the 3.5E Forgotten Realms Campaign Setting (page 78) and the 4E Forgotten Realms Campaign Guide (page 95). We can hypothesize that Greenest sees very little snow (but it might happen occasionally during particularly cold winters). It is therefore unlikely that they plant Winter Wheat due to the 30-60 day vernalization period it calls for at 0-5 degrees celsius. Also, it isn't uncommon for cultures with mild winters to plant Spring Wheat in the autumn (generally November or December) and then harvest it during the spring (generally April or May).
Since Greenest is in the process of milling grain into flour when the attack begins on the town (see page 11 of the module text, under the Save the Mill heading), they likely just recently harvested their local wheat crops for the growing season. The only remaining question that needs answered to approximate which month it is, is whether Greenest just harvested wheat grown over the summer or the winter. If they planted it in November and harvested it in May, then it is likely June by the time the module begins. If they planted it in April and harvested it in September, then it is likely October by the time the module begins.