I went and checked to confirm that bananas are grown in Iraq today, so it’s a safe bet that they could be grown there in Solomon’s time. The Hebrews didn’t have to go all the way to India.
And apparently Egypt grows them too, although it is the 56th largest exporter. Today the majority of their bananas go to Jordan and Palestine. I bet they could be picked green and get to Solomon’s temple in time for them to ripen.