Thought I would give you all an update.
Tinyportal was able to install today. Whatever was preventing it from installing before, clearly has *something* to do with my host.
The reason I say this is that all I did was wait 24 hours, then click apply mod again. This time, it installed without a hitch.
It did the same on all three of the sites I manage on that particular host.
If I get a definitive answer on this from my host, I will let you wall know. However, this is not the first time that something which refused to work, worked correctly after letting a night pass.