The development of methods to estimate the severity of Ulcerative Colitis (UC) is of significant importance. However, these methods often suffer from domain shifts caused by differences in imaging devices and clinical settings across hospitals. Although several domain adaptation methods have been proposed to address domain shift, they still struggle with the lack of supervision in the target domain or the high cost of annotation. To overcome these challenges, we propose a novel Weakly Supervised Domain Adaptation method that leverages patient-level diagnostic results, which are routinely recorded in UC diagnosis, as weak supervision in the target domain. The proposed method aligns class-wise distributions across domains using Shared Aggregation Tokens and a Max-Severity Triplet Loss, which leverages the characteristic that patient-level diagnoses are determined by the most severe region within each patient. Experimental results demonstrate that our method outperforms comparative DA approaches, improving UC severity estimation in a domain-shifted setting.