Measure low DOM absorbance values in a microplate reader. Is there a suitable deep well microplate with flat bottoms available?
I am not aware of suitable microplates for this application.  Here are some alternative approaches.

1. Low-volume quartz cuvettes.

There are quartz cuvettes in "micro" formats that may be suitable for you.  Mostly commonly they work with 1 cm pathlength.  The volume of the cells is controlled by the width of the non-optical dimension.  This can be made extremely small.  

One example I found lists cuvettes that still have path lengths of 1 cm with total fill volumes as low as 15 &mu;L.  

2. Sample concentration

Details depend heavily on the details of your DOM and application, but you could concentrate your samples and then use existing microplates.  

3. Longer measurement times.

Most plate readers (and cuvette-based spectrophotometers) let you control the number of "shots" or of the total amount of time a measurement of light intensity is integrated over.  You could crank this number up to get better sensitivity.