If transfering from Business Visa to Working Visa youre employer still have ample time but I dont think that you have enough time for the residence visa as you need to take the medical exam and the finger printing before you can have the residence permit or visa. If the case is that your visa will expire by the 20th of December its either you will renew/extend your business visa or you can exit on one of the GCC near Qatar but I dont think you can enter one of the GCC as your passport's validity is under 6 months (as most of the countries require at least 6 months validity of the passport). If its the case I think you need to do what the Philippine Embassy advised you.
If transfering from Business Visa to Working Visa youre employer still have ample time but I dont think that you have enough time for the residence visa as you need to take the medical exam and the finger printing before you can have the residence permit or visa. If the case is that your visa will expire by the 20th of December its either you will renew/extend your business visa or you can exit on one of the GCC near Qatar but I dont think you can enter one of the GCC as your passport's validity is under 6 months (as most of the countries require at least 6 months validity of the passport). If its the case I think you need to do what the Philippine Embassy advised you.