"Would it even be accepted by banks/government institutions?"

No. You can choose a Chinese name (or have one chosen for you) to make life easier for your Chinese friends and colleagues, but your bank account etc. will use your real name.

The name on your bank account is likely to be on of the below, depending on which bank you use, and what type of day the teller is having:

SURNAMEFORENAME

FORENAMESURNAME

FORENAMEMIDDLESURNAME

And you need to know what it is, because without the correct name it's impossible to receive inbound bank transfers, or to link your bank card with WeChat/AliPay. Your name is not usually printed on your bank card, which you are given as soon as you open the account.