Trade Name on Name Board

No. You can use name which has been registered with GST department.

Can I register new gst in the same pan? for the Branch shop.

For same state no need to apply new GST registration
if your business natural is vertical than you apply for new GST registration.

It is not mandatory to apply for new registration for branch shop. But there is not any restriction to apply for new registration for new branch shop. You can apply for new registration but it will unnecessarily increase your compliance burden.
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