In this letter, we present a custom-designed and flexible readout circuitry for the characterization of flexible and planar electrolyte-gated carbon nanotube field-effect transistor-based biosensors for ammonium detection in sweat. We employed spray-deposited semiconducting carbon nanotubes as active material, and functionalized the devices with previously synthesized ion-selective membrane, based on the nonactin ionophore. In the design of the readout circuitry, we focused on enabling low-power operation with a single coin-cell battery with compact wireless data transmission. To maximize the bendability of the flexible PCB, particular attention was taken in layout routing as well as in selecting small-sized packages for the commercial integrated circuits and all-in-one systems such as programmable Bluetooth system-on-chip (i.e., ST microelectronics BlueNRG SoC). We recorded a high sensitivity of 4.516 $\mu$A/decade for sweat ammonium level analysis between 0.1 and 100 mM, which fully covers the relevant range of interest in the context of sport monitoring. The characterization was carried out with the introduced front-end readout, and the results were benchmarked with a "gold standard" instrument, showing good and reliable performance of the developed fully flexible bioelectronic system.