We studied and compared single-side pumping (SSP) and double-side pumping (DSP) of a semiconductor membrane external-cavity surface-emitting laser (MECSEL). The MECSEL-active region was based on an AlGaAs quantum well structure embedded between two silicon carbide (SiC) wafer pieces that were used as transparent intra-cavity (IC) heat spreaders creating a symmetrical cooling environment. The gain structure targeted emission at 780 nm, a wavelength region that is important for many applications, and where the development of high-brightness high-power laser sources is gaining more momentum. By DSP at 20°C heat sink temperature, we could reduce the laser threshold from 0.79 to 0.69 W of absorbed pump power, while the maximum output power was increased from 3.13 to 3.22 W. The differential efficiency was improved from 31.9% to 34.4%, which represents a record value for SiC-cooled vertically emitting semiconductor lasers. The improvements are enabled by a reduced thermal resistance of the gain element by 9% compared to SSP. The beam quality was measured to be M2<1.09. Finally, we demonstrate a maximum tuning range from 767 to 811 nm. This wavelength range was not addressed by any MECSEL or vertical external-cavity surface-emitting laser device before and extends the available wavelengths for semiconductor based high-quality beam and high-power laser sources to a wavelength window relevant for quantum technology, spectroscopy, or medicine.