I'm answering this for future visitors. Here are the relevant details from the BAMF website
"If you are currently outside Germany, you should apply to the competent German mission abroad for a visa for the purpose of employment before you enter Germany. As a matter of principle, this also applies in cases in which visa-free entry would otherwise be possible. See the next paragraph for details of exceptions. Under no circumstances should you enter Germany with a tourist visa, as this type of visa can only be extended in the form of a residence title in exceptional cases. As a rule, it is otherwise necessary to leave and then re-enter the country. The visa for the purpose of employment entitles you to enter Germany and then to apply to the immigration authority that is competent for your place of residence for the EU Blue Card to be issued. You will find the contact details of the German missions abroad worldwide here:
Exceptions apply to nationals of Australia, Canada, Israel, Japan, the Republic of Korea, New Zealand and the United States of America. They can enter Germany on a visa-free basis and can apply to the competent immigration authority in Germany for their future place of residence for an EU Blue Card within three months of entering Germany."
Since you are not eligible for one of these exceptions, you cannot apply for your Blucard in Germany directly and you will need to visit one of the German Missions in your country.
HOWEVER, I think you should contact your future employer to see if they can do anything to help. You should also make an appointment with your consulate NOW so that you at least have that. You can always cancel it later if you find another means of getting the application in (I don't think you will, but not sure).
ETA: if you are eligible for one of the exceptions outlined above, your employer should be able to file all the relevant paperwork ahead of your arrival which will speed the process along. I'm American so I could get my permits after arriving, so my employer submitted my diplomas and work history, etc. and confirmed the approval for the work permit. Then I just went to the Ausländerbehörde in Munich, registered myself as a resident, went upstairs and finished the paperwork for the residence permit/work permit. I did indeed get the permits the same day, but that was because my application paperwork had already been reviewed and approved ahead of my arrival. I never had to visit the German Mission in the US.