Its hard when you are newly diagnosed because of the honeymoon effect.......but ultimately it is the insulin that is responsible for the lows......
the basal is the Levemir, yeah, so you could try removing that and see what happens......the only other option is to find something that you can stomach when the lows come......have you tried sugary juice, something like that which is quick acting and easily consumed...?
just make sure you keep testing so you have all the facts for your team to help..