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Breast cancer is a common and deadly disease, but it is often curable when diagnosed early. While most countries have large-scale screening programs, there is no consensus on a single globally accepted guideline for breast cancer screening. The complex nature of the disease; the limited availability of screening methods such as mammography, magnetic resonance imaging (MRI), and ultrasound; and public health policies all factor into the development of screening policies. Resource availability concerns necessitate the design of policies which conform to a budget, a problem which can be modelled as a constrained partially observable Markov decision process (CPOMDP). In this study, we propose a multi-objective CPOMDP model for breast cancer screening which allows for supplemental screening methods to accompany mammography. The model has two objectives: maximize the quality-adjusted life years (QALYs) and minimize lifetime breast cancer mortality risk (LBCMR). We identify the Pareto frontier of optimal solutions for average and high-risk patients at different budget levels, which can be used by decision-makers to set policies in practice. We find that the policies obtained by using a weighted objective are able to generate well-balanced QALYs and LBCMR values. In contrast, the single-objective models generally sacrifice a substantial amount in terms of QALYs/LBCMR for a minimal gain in LBCMR/QALYs. Additionally, our results show that, with the baseline cost values for supplemental screenings as well as the additional disutility that they incur, they are rarely recommended in CPOMDP policies, especially in a budget-constrained setting. A sensitivity analysis reveals the thresholds on cost and disutility values at which supplemental screenings become advantageous to prescribe.