Jeff zone 8 N.C.

The banana will do ok, but not to it's potential, with the grass. Grass robs a ton of nutrients, air and water from the roots of the banana plant. I would mow it and then cover with a mulch. Cardboard, laid over the mowed grass will suppress the grass and probably eventually kill it and not damage the banana roots. Maybe put a bark or straw (not hay/it has grass seed in it) mulch over the cardboard for a better look and to help the cardboard break down faster over time. If using smaller pieces of cardboard overlap them a lot, if the grass is a running type as the grass will try to find every crack in the cardboard to grow through.