The QuickBooks Online integration can map to Bank and/or Other Current Asset account types in QuickBooks. If you are attempting to map to an account that exists in QuickBooks but does not appear in Bloomerang, check QuickBooks to verify the account type. You can find a list of all your accounts in QuickBooks by clicking Accounting > Chart of Accounts. If the account is not a Bank and/or Other Current Asset account in QuickBooks, it won't be available for mapping via accounts in Bloomerang.
That said, service items in QuickBooks Online always connect to an income account. In Bloomerang, map to service items that are linked to income accounts to sync transactions that belong in these accounts. You can check your service items and which income accounts they are linked to in QuickBooks by clicking Sales > Products and Services.
We recommend consulting with your accountant or bookkeeper if you have any questions about mapping your transactions.