Can LawStream produce accounts that show a balance carried-forward from the previous account?

Yes, of course! That feature, and many others, are included in LawStream.
Because LawStream integrates timekeeping and billing and accounting data (along with various other types of data), you can count on LawStream to show accounting information along with other billing data.
LawStream’s bills have many other options, including an option to show past balances with new bills, and trust transfers (if applicable). That way, the carried-forward balance can appear on new bills as they are generated. You can also product statements in different formats, showing past billing and payments, apart from any new time and expenses that might appear on new bills.