GST,being a consumption based tax, if you see in case of imports from different country, place of supply had to be that other country and accordingly GST collected there. This is the main reason exports are considered as zero rated supply as government can't levy tax beyond its territorial boundry. So, your first proposal, stating "collecting tax from customer" is totally wrong both lawfully and concept wise as government can't charge from a person who is not even in the territory of India.
Secondly, then what does "Payment of IGST on export and claiming of refund later" as provided in law means:--- See, main concept of giving benefit of refund is that, government is giving option/ way to supplier to utilise his input tax credit, as if not given such option it would be impossible for exporters to use their credits and thereby it would be a loss to exporters which they would cover by raising price of their good which subequently would reduce exports.
So, 1st option given by government (LUT/Bond) is clear, that don't pay IGST, claim credit of your input tax as refund.
Now 2nd option (paying IGST on export) , it means pay applicable IGST on goods/service as applicable and claim refund of tax paid. Now, question is how to pay this IGST on export as you are saying does supplier has to pay from his pocket? So, here supplier can utilise his input credit to pay this output IGST and calim refund of tax paid now. If in situations IGST on export is more than credit available, then pay balance amount as cash after utilising credit and claim refund of entire tax paid.
Hope it is clear now!!