The efficiency of manual wheelchair propulsion is largely determined by rolling resistance, which depends on tire pressure, surface type, and wheel construction. A particularly significant source of energy loss arises from anti-rollback modules, where the contact between a rigid roller and an elastic tire generates an additional braking torque. The aim of this study was to experimentally determine the displacement of the pressure center (x0) within the contact zone between the roller of the anti-rollback module and the wheelchair tire, as a function of the normal load (F0) and the internal tire pressure (pt). The experiments were conducted using a custom-built test stand equipped with an HBM 1-T20WN torque transducer and a Zemic H3-C3 force sensor. The analysis included measurements of the braking torque (Tb) and the normal load across five pressure levels (3–8 bar) and eight loading conditions (25–200 N). The results showed that increasing tire pressure reduces tire deformation, enhances contact stiffness, and decreases the braking torque by approximately 10–15%. The displacement of the pressure center decreased with rising pressure: for F0 = 25 N, x0 decreased from 11.6 mm (3 bar) to 9.1 mm (8 bar), and for F0 = 200 N, from 18.5 mm to 16.0 mm. The linear model x0(F0, pt) achieved a determination coefficient of R2 = 0.957, confirming its strong agreement with experimental data. The findings indicate that maintaining tire pressure within the range of 6–8 bar minimizes rolling resistance torque, thereby reducing user effort and improving propulsion efficiency.