A laser wakefield accelerator (LWFA) with a weak focusing force is considered to seek improved beam quality in LWFA. We employ super-Gaussian laser pulses to generate the wakefield and study the behavior of the electron beam dynamics and synchrotron radiation arising from the transverse betatron oscillations through analysis and computation. We note that the super-Gaussian wakefields radically reduce the betatron oscillations and make the electron orbits mainly ballistic over a single stage. This feature permits to obtain small emittance and thus high luminosity, while still benefitting from the low-density operation of LWFA (Nakajima et al. 2011 Phys. Rev. ST Accel. Beams14, 091301), such as the reduced radiation loss, less number of stages, less beam instabilities, and less required wall plug power than in higher density regimes.