Moim zdaniem, chodzi o ewidencję bilansową tych aktywów (tu: środków trwałych lub wartości niematerialnych i prawnych), czyli ewidencję/rejestr (register) majątku trwałego (fixed assets).
'Balance sheet' to roczne zestawienie bilansowe (bilans, czyli zestawienie aktywów i pasywów) więc raczej nie chodzi tu o 'balance sheet register'.

A fixed asset register (FAR) is an accounting method used for major resources of a business.

Fixed assets are those such as land, machines, office equipment, buildings, patents, trademarks, copyrights, etc. held for the purpose of production of goods or rendering of services and are not held for the purpose of sale in the ordinary course of business.

http://en.wikipedia.org/wiki/Fixed_Assets_Register

A fixed asset register is used to ensure the correct information regarding fixed asset holdings is
recorded and entered into the accounting system.

http://www.homelearningcollege.com/HLC/media/PDF/SampleMater...

"Create a fixed asset register. Data typically captured in an asset register include:
An asset tracking number, which is a unique identification number. The physical asset is often marked with this identification number, either directly or with an asset tag.
An asset description.
The quantity of each asset.
The manufacturer of the asset.
...
Create a fixed asset account record for each new fixed asset."

http://www.wikihow.com/Prepare-an-Asset-Register

record in the balance sheet account(s)
record in one of the balance sheet accounts

http://www.usg.edu/business_procedures_manual/section2/C1104
...