Rename home directory without causing double Time Machine backup
I just tried enabling Time Machine on a VM, saving one snapshot, renaming the home directory, and saving another snapshot, and Time Machine did save a second copy of the whole home directory.

When I tried it again but used bypass to rename the home directory on the Time Machine volume before saving another snapshot, Time Machine did not save a second copy of the whole home directory.

So try running a command like this after you rename the home directory:

for f in /Volumes/*/Backups.backupdb/*/*/*/Users/oldname; do sudo /System/Library/Extensions/TMSafetyNet.kext/Helpers/bypass mv "$f" "${f%/*}/newname"; done