Kindly allow me to give you some preliminary line/s of defense:
'Existing law' - as per 2 (48) of respective state act/s - is defined as follows: “existing law” means any law, notification, order, rule or regulation relating to levy and collection of duty or tax or cess on goods or services or both passed or made before the commencement of this Act by the Legislature or any authority or person having the power to make such law, notification, order, rule or regulation;
Existing law' - as per 2 (48) of CGST Act, 2017 - is defined as follows: “existing law” means any law, notification, order, rule or regulation relating to levy and collection of duty or tax on goods or services or both passed or made before the commencement of this Act by Parliament or any Authority or person having the power to make such law, notification, order, rule or regulation;
M/s A was not liable to be registered under the 'existing law' - as defined under CGST Act - for its trading activities (i.e. as far as under laws by Parliaments are concerned).
He was registered under state act as 'trader' as per state legislation and that registration does not make him 'registered' under existing law under CGST Act, 2017.
Hence, for purpose of taking transition credit u/s 140 (3) (i.e. transferred as CGST) of the CGST Act, 2017, M/s A was eligible to the extent of his trading activities are concerned as he was not liable to be registered under the existing law.
Just because M/s A was registered under service tax related provisions for its 'business auxiliary services', it is not barred from taking benefit of 'proviso' to Section 140 (3) as it is not taking those benefit u/s 140 (3) in the capacity of service provider / manufacturer etc. but as a 'trader in goods'.
Considering that Section 140 are 'beneficial provisions' favoring tax-paper, when two interpretations of any such provision is possible, interpretation - which is beneficial to the tax-payer - needs to be adopted.
In other words, benefit under 140 (3) should not be denied to M/s A as 'trader in goods', just because it was registered under service tax in a completely unconnected / independent business activity.
As suggested by my colleague Shri Alkesh Jani Ji, I would also suggest to take assistance of expert to study whole issue deeper, including board clarifications etc., and have comprehensive analysis and get multiple line/s of defense before M/s A embark on defending itself on this complicated issue.
These are ex facie views of mine and the same should not be construed as professional advice / suggestion.