For fluorescence, you should use a black plate with black bottom, and use top fluorescence. This avoids the problem of the lack of transparency of the plastic to the excitation wavelengths. White plates should be avoided because they are fluorescent.

The previous answer only applies if you are not using microscopy assays. Otherwise, you definitely need bottom fluorescence and plates with transparent bottom.

In any case, I am surprised that high wavelength is the problem. Could it be that you read the plate description wrong and it is possible to use the plates for wavelength range OVER 340 nm? It is UV range that is tricky, not VIS. Hence, you may have problems with excitation at 300 nm, not with emission at over 400 nm.

If your excitation or emission light goes through your vessel, be it cuvette, multiwell plate, or any other, the vessel needs to be transparent at both excitation and emission wavelengths. Your plate essentially acts as a 340 nm long-pass filter. Probably it is not awfully good as a filter, so you might be willing to suffer losses near the 340 nm limit, but it does cut out most of your excitation when you go lower wavelengths. The problem with plastics is that they are mostly poorly transparent in the UV region of the spectrum or absorb biological compounds. Quartz microplates of course tend to be more expensive, if you need to work near 300 nm wavelength.

FYI