I've finally gotten this to do it again, and actually had time to look at it in detail.
The program is started automatically at boot by /etc/rc.local
At startup time, the program now saves in globals:
  AppStrtTime = time();
  AppStrtBasetime = $^T;
  UnixUpAtAppStrt = Unix::Uptime->uptime();

The next afternoon I query the program and get:
  System uptime: (81700) 0d 22:41:40
  Program uptime: (82550) 0d 22:55:50
  UnixUpAtAppStrt: 27
  AppStrtTime: 1684095442
  AppStartBasetime: 1684095441
  time() now: 1684177991
  ^T now: 1684095441

Armed with these numbers, reality check:
  UnixUpAtAppStrt = 27 -- sounds reasonable - boot to Perl app startup took half minute
  AppStrtTime follows within a second of AppStrtBasetime -- seems right as ^T was set when the app started, which would be before the code in the app actually started to execute and perform the time() call
  ^T now -- matches the ^T from startup, it has not changed

With AppStrtTime time, and assuming UnixUpAtAppStrt was correct, the actual system startup time was:
  1684095442 - 27 = 1684095415
Subtracting that from time now():
  1684177991 - 1684095415 = 82576
yet:
  Unix::Uptime->uptime() = 81700
WRONG! This should be 82576.
If it was only a few seconds off, I wouldn't have even noticed, but it's off by 14+ mins in under 24 hours!

Maybe some day I'll have a look at Unix::Uptime, but it may be a while - like years.