In reference to your query, you are providing consulting services to an entity located outside India, and accordingly the same would be classified as export of services subject to satisfaction of the specified conditions.
Your transaction will be reportable as zero rated transactions, and subject to submission of LUT you would not be required to pay taxes on your service exports. However if LUT is not submitted, then in that case you would need to deposit GST and later on claim refund of the same.
It would be advisable for you to file LUT and proceed without payment as it involves less hassles as also saves on cash flow and compliance cost for claim of refund.
There would be no problem to you in your remittance. Your Italian client getting their production done in India is irrelevant for you, as that is an independent transaction and not connected to you but placed with a different supplier.
Please advise in case of any further clarification.
Thanks & Regards,
CA Aditya Dhanuka.