the lats play a very minor role in bench. the origin and attachment of the muscle belly(the anatomical structure of the lats) doesnt allow for the lats to take any part in the lifting of the weight. the lats only contribute a very small amount of ancillary support and stabilization.
your bench wont go up by training your lats alone. provide one example of this ever happening, or provide some proof that strengthening the posterior chain could somehow improve the strength capacity of the anterior (to a significant degree).
in order for your bench to increase first and foremost you need to increase the strength of your chest. increases in deltoid and tricep strength can also increase yoru bench, but not anywhere near the degree that chest strength can, assuming proper form (majority of the weight is in the pecs). outside of those three prime movers no other muscle is going to be able to contribute much at all, if anything, to bench press. the lats, being directly opposite the chest, can play no part whatsoever in lifting of the bar during bench.