Online Trading Account Opening and Existing DMAT Account

I have a DMAT account with a nationalised bank which does not provide online share trading account. Is there any service provider who can provide an online share trading account but link it to the DMAT account of this nationalised bank?

I am an NRI staying overseas in Sydney, Australia. My primary purpose in opening this trading account is to dispose off shares I have held for a long time now (NRI non-repatriable --- shares obtained as Resident Indian). Since I am overseas I want to be able to open this account remotely and do further trading in as paperless a manner as possible.

If I open a 3 in 1 account would I be able to transfer the shares in the existing DMAT account with the nationalised bank to the new DMAT account with any issues?

Please suggest best solution and some service providers.

Thanks & Reagrds