@Wrapmate Welcome to the Netlify community.
I thought I was following you until that last paragraph. If you want to keep Route 53, then get SiteB ready and make the DNS changes with Route 53 when you’re ready. If you want to delegate DNS to Netlify, then someone will have to transfer most of those 114 DNS entries from Route 53 to Netlify and then when ready, change your DNS delegation. The big problem is that your biggest disruption will come from switching to SiteB, and THEN changing its domain name.
At any rate, I refer you to this source: