Preservation of the femtosecond (fs) microbunches, created during laser acceleration, is a crucial step to enable staging of the laser acceleration process. This paper focuses on the optimization of the beam dynamics of fs microbunches transported through the staged electron laser acceleration (STELLA-II) experiment being carried out at the Brookhaven National Laboratory Accelerator Test Facility. STELLA-II consists of an inverse free electron laser (IFEL) untapered undulator, which acts as an electron beam energy modulator; a magnetic chicane, which acts as a buncher; a second IFEL tapered undulator, which acts as an accelerator; and a dipole, which serves as an energy spectrometer. When the energy-modulated macrobunch traverses through the chicane and a short drift space, microbunches of order fs in duration (i.e., $\ensuremath{\sim}3\text{ }\text{ }\mathrm{fs}$ FWHM) are formed. The 3-fs microbunches are accelerated by interacting with a high-power ${\mathrm{CO}}_{2}$ laser beam in the following tapered undulator. These extremely short microbunches may experience significant space charge and coherent synchrotron radiation effects when traversing the STELLA-II transport line. These effects are analyzed and the safe operating conditions are determined. With less than 0.5-pC microbunch charge, both microbunch debunching and emittance growth are negligible, and the energy-spread increase is less than 5%. These results are also useful for the laser electron acceleration project at SLAC and in possible future programs where the fs microbunches are employed for other purposes.