Gst registration for different business verticals

Dear Members,

If a proprietor having more then one business (which trade name & nature is completely different) in one state and register himself in GST then in registration there is only one Trade Name option available by this he left with 2 options,

1. Either register his 2 business seprately under different business verticles & increase their compliance cost,or
2. Register himself with one registration add HSN/SAC of his products/services alongwith addition of business places.

Here question aries if he opt for option two then how will he claim ITC on his second business (which trade name is not registerd in GST) becuase if he take purcahse bills in the name of his second business then there will be "Customer Name: mismatch while counter party will file GSTR1 beacuse in this GSTIN only first tradename is registered.How to overcome with this situtation.(Ex. his first business is Hotel & 2nd one is Gym, Hotel business trade name registerd but he want to purchase machinery for Gym. In which name & GSTIN bill should be obtain)

Please assist.