Is it possible to calculating sea-level pressure with an unknown base station altitude?
Calculating sea level pressure depends on knowing your altitude.  A function that does this is the hypsometric equation $$\Delta z = \dfrac{R \overline{T}}{g}\ln\dfrac{p_1}{p_2}.$$  This function gives the depth ($\Delta z$) of an atomspheric layer with average temperature $\overline{T}$ between pressure levels $p_1$ and $p_2$.  In your case, the station pressure is known and the height of mean sea level is known; in order to calculate your height or to calculate sea level pressure, you must know the other quantity.  The average temperature is also an unknown, but you can approximate this or use a value based on the lapse rate of the standard atmosphere.