So this guitar is quite a unique model, the John Lennon signature. Its pickup is not an acoustic-guitar pickup at all but, well, basically a standard high-impedance humbucker in small format.
It's no big surprise then that it sounds more like an electric guitar: such a pickup, together with the cable capacitance, forms a 2nd order lowpass filter, and that gives the characteristic electric-guitar sound. (Why did you buy such a guitar if you don't want its characteristic sound??)
To get a more acoustic sound, the best option is of course to use a microphone. Unfortunately that brings lots of problems in a live setting. Magnetic pickups actually remain the best in terms of feedback robustness etc.. They can also give a decent acoustic sound, but it's crucial that you avoid the resonant lowpass-filter effect. Properly designed modern acoustic PUs achieve this through a low-impedance spec, i.e. the coil uses thicker wire with fewer windings, which leads too a much smaller inductance. Unfortunately it also makes the signal voltage much weaker, but that can actually be compensated easily with a preamp.
If you want to keep the PU as it is in the guitar, there's another parameter you can change to keep the filter in check: you can lower the capacitance. It's mostly the cable that's responsible for this, so you want a decoupling circuit right after the pickup. This can be a very simple buffer:
(created with CircuitLab)
After that, the frequency response should be much clearer.
Of course make sure you don't use an electric-guitar amp but an acoustic amp or go straight into the PA via a DI, best an acoustic-optimised one like a Radial Tonebone or LRBaggs ParaDI. But with the active circuit in your guitar, a cheap passive DI like the Palmer PAN01 should also be fine.