In terms of direct numbers at present, dark energy comprises about 70% of all of the energy in the universe. Radiation, on the other hand, makes up less than 0.005% of the energy in the universe. It's such a small fraction that it's less than the error associated with the values for matter and dark energy.
A good way to approximate how the two energies compare over time (with expansion included) is through the scale factor of the metric, $a$. The scale factor represents the ratio of the distance between two points at any given moment of time to the distance between those two points now. Naturally, as the universe expands, the amount of volume in a given region of the universe increases like $a^3$. With that said, let's take a look at the volume density of both your types of energy.
As you quite correctly pointed out, the expansion of the universe redshifts radiation, which means the universe loses that energy entirely. Add to that the fact that the number density of a fixed number of photons is proportional to $\frac{1}{a^3}$, and it's not hard to understand why relativity says the total energy density of radiation decreases like $\frac{1}{a^4}$. In other words the total amount of energy contained in radiation decreases approximately like $\frac{1}{a}$.
As for dark energy, the current accepted model, $\Lambda$-CDM, treats dark energy like a constant energy density. That means as the universe expands, the amount of dark energy per unit volume remains constant. Yikes. This means the total amount of dark energy increases like $a^3$.
Add them together and you see there is a net increase in total energy of the universe (the total energy of matter remains more or less constant). Clearly it isn't the case that the energy lost from radiation is picked up as dark energy. But, of course, you already knew that. You had already gone as far as realizing that the radiation energy fell off like $\frac{1}{a}$ and there would need to be a seriously funky (<-- technical term) relationship between $a$ and the energy density of dark energy for the two totals to sum to a constant. Kudos to you for having figured this out by yourself and asking an excellent follow-up question.