This is how I analyze the case you describe.
Windows Explorer is strangely programmed, in the sense that there is always only
one explorer.exe instance, even if several Explorer windows are opened,
and it is started during logon as a standard user, even if the login account
is that of an administrator.
This means that Explorer never runs in elevated mode, and that any process
started via Explorer inherits this running mode. If the started process wants
to acquire administrator permissions, it must demand elevation, thus requiring
user approval (if UAC is enabled).
On the other hand, if a process is started from an elevated cmd.exe,
it inherits this running mode, so it does not need to request elevation
(and if it does then this is immediately "granted" with no UAC prompt).
Now, I have started in the past installation programs from Explorer so many
times, and they always requested elevation and worked without fault.
To be sure of the facts, I started an elevated cmd.exe as Admin both
from the command-line and from Explorer and compared their permissions using
Process Explorer.
All permissions for both cases were absolutely identical.
This means that if the installation program you used was correctly programmed,
it would have had no problem. The inescapable conclusion is that it was
badly and weirdly programmed.
The fact that it found it necessary to rename its own folder, is already
an indication of something really strange.
My guess would be that it was created by one programmer, then was added-to
by another programmer that didn't dare modify the original code and only
appended more stuff to it.
I have tried to imagine what kind of error could cause such a problem,
and I only found the following, which you can verify by watching
the installation as it's working, using Process Explorer as the tool.
The installation is not one process, meaning that the initial setup
launched more than one process. Perhaps not all these processes requested
elevation.
That folder was deliberately created with permissions that precluded
a non-administrator from modifying it, and the rename was attempted by
an invoked rename process that wasn't elevated
(it would have been elevated if it were launched from an already elevated process).
You may verify this by right-click of the folder in Explorer,
choosing Properties, and examining the Security tab.
The rename was attempted by one non-elevated process while the folder was
still opened by another elevated process, so couldn't be renamed.
These are not the only possibilities - there is an unlimited number
of possible programming errors.
The problem can only be fully understood by carefully
observing the installation while it is working.