High-intensity X-ray sources are invaluable tools, enabling experiments at the forefront of our understanding of materials science, chemistry, biology, and physics. Laser-plasma electron accelerators are sources of high-intensity X-rays, as electrons accelerated in wakefields emit short-wavelength radiation due to betatron oscillations. While applications such as phasecontrast imaging with these betatron sources have already been demonstrated, others would require higher photon number and would benefit from increased tunability. In this paper we demonstrate, through detailed 3D simulations, a novel configuration for a laser-wakefield betatron source that increases the energy of the X-ray emission and also provides increased flexibility in the tuning of the X-ray photon energy. This is made by combining two Laguerre-Gaussian pulses with non-zero net orbital angular momentum, leading to a rotation of the intensity pattern, and hence, of the driven wakefields. The helical motion driven by the laser rotation is found to dominate the radiation emission, rather than the betatron oscillations. Moreover, the radius of this helical motion can be controlled through the laser spot size and orbital angular momentum indexes, meaning that the radiation can be tuned fully independently of the plasma parameters.