Wheat production and end-use quality are severely threatened by drought and heat stresses. This study evaluated stress impacts on phenotypic and gluten protein characteristics of eight spring wheat genotypes (Diskett, Happy, Bumble, SW1, SW2, SW3, SW4, and SW5) grown to maturity under controlled conditions (Biotron) using RGB imaging and size-exclusion high-performance liquid chromatography (SE-HPLC). Among the stress treatments compared, combined heat–drought stress had the most severe negative impacts on biomass (real and digital), grain yield, and thousand kernel weight. Conversely, it had a positive effect on most gluten parameters evaluated by SE-HPLC and resulted in a positive correlation between spike traits and gluten strength, expressed as unextractable gluten polymer (%UPP) and large monomeric protein (%LUMP). The best performing genotypes in terms of stability were Happy, Diskett, SW1, and SW2, which should be further explored as attractive breeding material for developing climate-resistant genotypes with improved bread-making quality. RGB imaging in combination with gluten protein screening by SE-HPLC could thus be a valuable approach for identifying climate stress–tolerant wheat genotypes.