It depends on the nature of the spectral transition. If the transition arises strictly between two electronic states, then (removing instrument from the equation) I can't think of a mechanism whereby the peak maximum would have a different Beer-Lambert behavior than an off-center wavelength. Of course most peaks in the UV-Vis spectrum are superpositions of multiple transitions, and the lambda max may not even be the maximum of any of the individual components. Different transitions can in principle have differing nonlinear absorption effects, although most UV-Vis experiments I wouldn't imagine use a powerful enough light source to really observe these effects strongly. (Similarly, concentration can affect different transitions to different degrees - much more likely to be observed in a UV-Vis experiment.)
Regarding the monochromaticity, the important condition is not the absolute monochromaticity per se but the spectral bandwidth of the incident light versus that of the spectral transition. If the bandwidth of the incident light is very broad, and you excite on the slope of the absorption band where the extinction coefficient varies considerably as a function of small changes in wavelength, you can inaccurate measurements of absorptivity (e.g., extinction coefficient) because of nonuniform absorption of different incident light wavelengths - particularly at high concentrations. For this reason it is best to measure the extinction coefficient at the peak maximum as a general rule. But honestly for most good UV-Vis instruments I wouldn't say that this is not a huge problem because of (1) the broadness of most electronic transitions and (2) the standard practice for measuring extinction coefficients is to use low concentration and narrow excitation slits so that you are most likely to be in the range where the B-L law applies. The exception would of course be very narrow transitions, such as some lanthanide complexes, maybe. But I guess if you want to be technical, then yes the Beer-Lambert Law fails to a greater degree as you excite farther away from the areas of the spectra that have constant absorptivity, which is most cases is far away from the lambda max value.