I am not familiar with Qubes OS but it looks like there is a common TemplateVM and apps are using separated AppVMs which share the root filesystem with respective TemplateVM. An AppVM has only read-only access to the filesystem of the TemplateVM on which it is based, so therefore an AppVM cannot modify a TemplateVM in any way. So maybe that is the reason why you can write into your home directory but not into system directories.
The issue then becomes that the Template has internet access restricted to the URL's for the enabled repositories for yum, so using the TemplateVM to write the blockchain updates won't work.
Maybe the solution is to split the datadir; have the wallet file placed in the read-only part of the filesystem when using the Template, and then have the blocks stored in the default home directory. I'm not sure bitcoin caters for a split datadir though, I will take a look at this.
Edit: this wouldn't work either, the wallet file still needs to be written to. Qubes maybe needs a dedicated build of Bitcoin, for now I'm contemplating just using the home directory as per Debian/Ubuntu.