Should we use modulus switching when dealing with ciphertexts encrypted in different rings?
The problem is that operations between such $c_1$ and $c_2$ are not even defined. You have to reduce the coefficients of the resulting polynomial modulus either $q_2$ or $q_4$, so, you have to switch one of them.

Switch $c_2$ into $R_{q_2}$ would work as well, but since it would increase the noise in $c_2$, it is smarter to switch $c_1$ into $R_{q_4}$ (which gives you meaningful operations and also decrease the $c_1$'s noise).

I think the only way to avoid this switching is to treat both the polynomials as elements of $R$ and check if the coefficients of the resultant polynomials are not greater than $q_4$, because this means that no reduction would be done in the coefficients if operations were done in  $R_{q_4}$ and so the resultant polynomial is a valid ciphertext in $R_{q_4}$ and in $R_{q_2}$.

Anyway, the modulus switching operations is not expensive, and it also reduces the noise, so, I don't see advantages in avoiding to do them.