Is it appropriate to compare the coefficient of variation between discrete (e.g., clutch size) and continuous data (e.g., egg width)?
There is a great variety of the different discrete variables: nominal, ordinal, and count. Gender is nominal: you cannot attach a meaningful number to gender. Strongly disagree/disagree/agree/strongly agree is ordinal: they represent discretization of an underlying spectrum. You can code them as 0/1/2/3, or 1/2/3/4, or -2/-1/1/2, with an understanding that higher values of the coded variable represent the greater strength of agreement, although the difference between 1 and 2 may not be the same as the difference between 2 and 3. Clutch size is a count variable: the value of 1 is meaningful, and the differences between 2 and 3 is the same difference as between 8 and 9. For most purposes, that works pretty much the same way as measuring egg width: the difference of 1cm between 1.5 cm and 2.5 cm is the same as between 3.2 cm and 4.2 cm. For both clutch size and egg width, the value of zero (nothing) is meaningful. So I would not see any objections to at least computing the CV for either variable.
Now, whether the substantive interpretation you offer would hold water should be an argument. For one thing, you have not convinced me! You would need to find sufficiently different nesting conditions to argue that a species reacts to the change in the environment by increasing CV of one or the other variable -- or, if you at the liberty of doing so, set up an experiment with whatever conditions are appropriate (dry vs. wet, warm vs. cold, plenty fish vs. no fish -- you know better.)