Depending on your use-case and the sensitivity of your situation installing Tor in C:\Temp is an okay alternative. Some very mild and rather theoretical issues I can think of is that IG C:\Temp is a less “secure” folder then c:\Programs where the latter has more strict file permissions. etc.
And note that C:\Temp is - like the name suggests - intended for temporary files and it is possible software like disk space optimizers might wipe the content of that folder from time to time. If you are aware of this, it’s ok.
As an addition, I also found this post discussing a problem similar to yours giving some more suggestions: https://tor.stackexchange.com/questions/3228/windows-installation-location
TL;DR: The C:\Temp solution is okay and Tor will work fine. But I just want to point out that it’s not the most optimal IMHO.
Depending on your use-case you might also be interested in Tails. It might look more complicated from the start, but in the end actually abstracts a lot of risks and technical considerations in the usage of Tor for most users.