The performance of alpha-type Stirling cryocoolers is strongly influenced by both thermodynamic and mechanical design parameters. This paper investigates the effect of the bore-to-stroke ratio on the cooling output of a cryogenic Stirling device while maintaining a constant swept volume of the compression space. An isothermal thermodynamic model was employed to evaluate pressure variations and cooling output for different piston diameters and head clearances. The results show that increasing the bore-to-stroke ratio reduces cooling output because of the associated increase in dead volume. However, for head clearances below 0.15 mm, this effect becomes significantly less pronounced, allowing the use of larger piston diameters and shorter strokes to reduce the overall dimensions and mass of the device. The influence of the bore-to-stroke ratio on piston side loading was also analysed by evaluating the normal force acting on the piston guiding section. Although larger bore-to-stroke ratios improve the durability of the piston-cylinder pair by reducing surface pressure, they simultaneously decrease cooling performance, indicating the existence of an optimal design compromise. Furthermore, crankshaft axis eccentricity and a balance-shaft system were investigated as methods for reducing piston side forces. A crankshaft axis eccentricity of −0.2 mm combined with second-order balance shafts reduced the normal-force amplitude by up to 63% compared with the conventional mechanism. The results indicate that an appropriate combination of geometric and crank-mechanism parameters can provide a favourable compromise between thermodynamic performance, piston side loading, and vibration-related forces.