Do you need new cables because the old ones no longer reach? If so, get the same cables but longer
Now, I'm thinking I'll probably go with 12 akg BlueJeans, primarily because I have been surrounding by SoundKing-looking wires for a few decades now.
That is not a good reason to switch cable brands, unless you just want better looking or different looking cables and they are visible all the time.
My question though, is I have a "bright" receiver (Yamaha RX-V795) with "bright" Infinitys. Other than sticking with copper over anything silver (which I doubt I would spring for at this point anyways), is there anything I should consider for avoiding a potentially harsh sounding system?
Firstly, the Yamaha is not a bright receiver. Cannot speak about the speakers response, but your room has a whole lot to do with sound and no cable will cure that problem. There is absolutely nothing wrong with copper cables, ZERO. Has nothing to do with sound of cables that do not exist.
You need to do something with your rooms acoustics to solve your problems.
I just don't want to be wasting money if I do. Thanks.
You would be wasting with more expensive cables that cannot solve your problems
Spend your money on room acoustics, or even different speakers on top of the room acoustic fix. You may need both fixed.