They cannot pay GST exclusive amount. If you are registered for GST, you have to charge and collect GST. It’s important that you collect it because you will have to pay that amount to government.
Your customer will claim GST paid to you as a credit (so they will get GST refund or it will be offset against GST they collect from their customers). I think you are both confused how GST works. There is no such a thing as “we are responsible for own gst”
Your other option, if your gross income is less than $75,000 per year, you can unregister from GST. Then you don’t have to charge it and your customer doesn’t have to pay it.