This sounds like every organization I've managed projects in and every company that I've consulted for along the way.
If this is your organization, too, then you likely need the best of the best in terms of a business analyst on each project being led and executed on for the project customers your organization is serving.
If you do need the “best of the best”, then what skills or characteristics are you looking for? What defines the best for your organization's project needs?
While BAs are not project managers, the most successful BAs manage the entire business analysis effort. This means that the BA is proactive and dependency-aware. It also means they manage themselves well, the stay on track with respect to commitments and deadlines, and can handle task delegation, decision-making, and issue management as needed on the project...