High-dimensional states (HD) are promising for quantum key distribution (QKD) due to their noise tolerance and efficiency. However, creating, and measuring, HD states is technologically challenging, thus making it important to study HD-QKD protocols where Alice and Bob are restricted in their quantum capabilities. In this paper, we revisit an HD-QKD protocol, introduced in (PRA 97 (4):042347), which does not require Alice and Bob to be capable of sending and measuring in full mutually unbiased bases.   The previous proof of security for this protocol has relied on numerical methods.  In this work, we provide a new proof of security, enabling the key-rate evaluation beyond previous work. Furthermore, our new proof produces better results for certain channel and dimension scenarios than prior work.