What is the purpose of the projected solid angle $dA^\perp$ term in the definition of Radiance? In the definition of Radiance, which according to PBRT is defined as


  flux per unit solid angle $d\omega$ per unit projected area $dA^\perp$


$$L_i=\frac{d\Phi}{d\omega\ dA^\perp}$$

My original assumption was that the $dA^\perp$ term is used to weaken the irradiance contribution due to the incident angle (which I assume is just Lambert's cosine law). However, in the rendering equation, we've already explicitly apply Lambert's law with the $\cos\theta$ term:
$$\int_\Omega L_i(\omega_{i}) \, f_r(\omega_{i}\rightarrow \omega_{o}) \, \cos\theta_{i} \, d\omega_{i}$$

I assume we arn't applying Lambert's law twice, so what exactly is this $dA^\perp$ term?
Your main idea is more or less correct. The cosine hidden in the projected area measure $dA^\perp = dA\cos(θ)$ compensates the weakening of irradiance due to incident angle (the Lambert's cosine law). This makes radiance independent from the incident angle. My guess is that the main motivation was to make it more practical to work with.

The cosine in the rendering equation comes from the definition of BRDF:

$$
f_{r}\left(\omega_{i}\rightarrow\omega_{o}\right) =\frac{\mathrm{d}L_{o}\left(\omega_{o}\right)}{\mathrm{d}E\left(\omega_{i}\right)} =\frac{\mathrm{d}L_{o}\left(\omega_{o}\right)}{L_{i}\left(\omega_{i}\right)\mathrm{d}\sigma^{\bot}\left(\omega_{i}\right)} =\frac{\mathrm{d}L_{o}\left(\omega_{o}\right)}{L_{i}\left(\omega_{i}\right)\cos\theta_{i}\mathrm{d}\omega_{i}}
$$

Which can be rewritten as

$$
\mathrm{d}L_{o}\left(\omega_{o}\right)
=f_{r}\left(\omega_{i}\rightarrow\omega_{o}\right) L_{i}\left(\omega_{i}\right) \cos\theta_{i} \mathrm{d}\omega_{i}
$$

Which can be integrated over the hemisphere the get the radiance reflected in a given direction $\omega_o$

$$
\int_\Omega f_{r}\left(\omega_{i}\rightarrow\omega_{o}\right) L_{i}\left(\omega_{i}\right) \cos\theta_{i} \mathrm{d}\omega_{i}
$$

Voila, the rendering equation!

Conclusion: Althought both cosines come from the same fundamental priciple, they serve different purposes.