Either the math and physics doctoral students at Jawaharlal Nehru University wrong or I was wrong.
Below is my explanation of the (Mc)-Term:
___________
To understand the trick, one must first observe the frequency
equation derived in the lecture:    • Lecture to Physics Ph.D Students | Ja...  

Mc^2 =(Mc)^2/M

Clearly, if we let M → ∞ without using the trick, the second term in the frequency will diverge. This is manifestly unphysical. Why must we let M → ∞ in the first place? The photon has no mass, and the mirror has some finite mass, so it is a valid approximation to let M → ∞. Furthermore, consider the
following thought experiment:
1. The mirror is not inclined, and thus standing vertical (θ′ = 90)
2. The angle of incidence, and thus reflection are both 0 (α = β = 0)

This is clearly wrong! We find later in the derivation that θ' = θ and ∆v → 0, which implies that f′ = 0. Clearly, the frequency of the reflected light beam cannot be 0! That would violate conservation of energy! That means that this second term must approach 0 as the mass of the mirror approaches ∞ based on physical grounds. This is the motivation behind the trick in 1. One may object to this trick on mathematical grounds, but is completely reasonable on physical grounds. In fact, Gjurchinovski, who re-derived Einstein’s Equation for Relativistic Reflection, used this exact same trick to

Our approximation that M → ∞ implies that ∆v → 0. If one doubts the validity of the derivation above, there is a completely different approach which will result in the exact same equation: in the frame of
the mirror, it is at rest. Thus, the standard law of reflection applies. One may then use the lorentz transformations to take the reflection light ray and find its lorentz-transformed counterpart in the frame in which the mirror is moving. In fact, this was Einstein’s derivation of the relativistic reflection equation (but for a vertical mirror). It must be emphasized that both approaches will result in
the same reflection law.

This concludes the explanation of the diverging mass trick.