Using stock tools, I couldn't see an easy way to transform multiple columns simultaneously. Still, one could write out three separate files. See: https://github.com/snodnipper/bng-transformations/tree/master/examples/ogr2ogr

2.vrt

<OGRVRTDataSource>
    <OGRVRTLayer name="process">
        <SrcDataSource>process.csv</SrcDataSource>
        <GeometryType>wkbPoint</GeometryType>
        <LayerSRS>+proj=tmerc +lat_0=49 +lon_0=-2 +k=0.9996012717 +x_0=400000 +y_0=-100000 +ellps=airy +datum=OSGB36 +units=m +no_defs +nadgrids=/Users/osnowden/code/bng-transformations/OSTN02_NTv2.gsb</LayerSRS>
        <GeometryField encoding="PointFromColumns" x="MBR_XMIN" y="MBR_YMIN"/>
    </OGRVRTLayer>
</OGRVRTDataSource>

3.vrt

<OGRVRTDataSource>
    <OGRVRTLayer name="input">
        <SrcDataSource>input.csv</SrcDataSource>
        <GeometryType>wkbPoint</GeometryType>
        <LayerSRS>+proj=tmerc +lat_0=49 +lon_0=-2 +k=0.9996012717 +x_0=400000 +y_0=-100000 +ellps=airy +datum=OSGB36 +units=m +no_defs +nadgrids=/Users/osnowden/code/bng-transformations/OSTN02_NTv2.gsb</LayerSRS>
        <GeometryField encoding="PointFromColumns" x="MBR_XMAX" y="MBR_YMAX"/>
    </OGRVRTLayer>
</OGRVRTDataSource>

input.vrt

<OGRVRTDataSource>
    <OGRVRTLayer name="process2">
        <SrcDataSource>process2.csv</SrcDataSource>
        <GeometryType>wkbPoint</GeometryType>
        <LayerSRS>+proj=tmerc +lat_0=49 +lon_0=-2 +k=0.9996012717 +x_0=400000 +y_0=-100000 +ellps=airy +datum=OSGB36 +units=m +no_defs +nadgrids=/Users/osnowden/code/bng-transformations/OSTN02_NTv2.gsb</LayerSRS>
        <GeometryField encoding="PointFromColumns" x="GEOMETRY_X" y="GEOMETRY_Y"/>
    </OGRVRTLayer>
</OGRVRTDataSource>

run.sh

ogr2ogr -f 'CSV' -lco GEOMETRY=AS_XY -t_srs EPSG:4326 -nln output process.csv 3.vrt
ogr2ogr -f 'CSV' -lco GEOMETRY=AS_XY -t_srs EPSG:4326 -nln process process2.csv 2.vrt
ogr2ogr -f 'CSV' -lco GEOMETRY=AS_XY -t_srs EPSG:4326 -nln output output.csv input.vrt