Although I am speaking with merely a budding technical knowledge base, I would point out that off-axis movement with mics commonly used for vocals can also impact sound-level consistency. Increased loss of sound level occurs with greater movement left/right or up/ down away from on axis, practically speaking the center of a mic. While some singers use a mic off axis intermittently for desired effects (as well as intentionally varying distance), it may be an unintentional occurrence that becomes noticeable in listening to a vocal track as opposed to awareness of this occurring on stage.
It occurred to me that a pop-filter screen (if not part of a shock-mount with your mic) with a flexible gooseneck and clip may give the singer a good reference point of where to be (in distance as well as centeredness).
It may be, however, that the singer’s vocal production is a variable that needs to be considered. After addressing technical consideration to no avail, if you know a trained singer who could discretely observe the singer to determine such an issue, it might be practical before suggesting a voice coach. If a coach is needed, a singing instructor at a community college might give private instruction. This would be preferably an instructor who has experience in using a mic to lend of insight as well as instruction in vocal production.
By the way, I did not see the mic itself discussed. If the sound level variation is occurring consistently within a defined frequency area or areas, I would suggest a test swap to a better vocal mic--borrowed if need be--with a flatter response to isolate whether the current mic is producing the sound level variations. Although perhaps not the only issue, lower priced mics commonly have noticeable response deviations due to their lack of engineered flatness. (As a technical term, flatness is used to refer to frequency response—not tonal quality—of mics’ or speakers’ relative accuracy, top to bottom in frequencies either captured or reproduced of sound, respectively. Generally speaking, greater flatness is desirable.)
Welcome to your Home Recording forum—JeffF.