Would it be possible to change the border texture for the clock, similarly to how we can already change the border of the minimap itself?

Since basically no one would use such a feature, it's unlikely.

Wow, that was a bit harsh. Is "no one" = yourself? Guess I'll implement it in a fork of my own then

No one as in, nobody has requested such a feature in 15 years, so I cannot personally put the time in to work on it. If you know Lua and are willing to work on it, pull requests for this addon are open.