Did exactly this kind of rebrand migration last year and underestimated how complex running both streams simultaneously actually gets. The technical debt from the old platform kept interfering with new feature work and it became a real bottleneck. A colleague pointed me toward enterprise cybersecurity consulting  and what worked well was that they structured the migration and new buildout as parallel tracks with clear handoff points. Kept things from bleeding into each other. If you need a team that can hold both workstreams without dropping either one it's worth reaching out.