So, I saw your post when it was fresh, been asking around, and it took a bit of time to get replies. Manufacturers I deal with have lots of great HDMI over Cat5 solutions. None of them deal with audio return channel. The reason I got was they are going for full implementation of HDMI (whatever that means if you leave off ARC), and I guess you have to give something up to get ARC. I didn't dig in deeper for a better understanding, sorry, but it seems a valid explanation.
If you google "audio return channel over cat5" and drill down a bit, you'll find a couple of solutions. It's possible to do, there are compromises, but it's all probably better than trying to fish a new cable. At least, worth a try.
If it helps at some point, take a look for "flat HDMI cable", and see the offerings there. I don't know if you could fit a relatively thin, flat cable in somewhere, under a carpet edge, behind a base board, or something, to solve your issue. I don't think it's cheaper, another way to go possibly.
Many TVs have optical out for digital audio. Optical cables are often very thin, and might also more easily hidden, and certainly cheaper. I've done this for folks with older TVs pre-ARC, and it works fine, and it's 5.1 capable.
One other possibility is a back-haul of audio only via a wireless audio link like this:
Audioengine W3 - Audioengineusa.com
The compromise there is, it's only two channel, not 5.1 digital. But sort of affordable, and the audio quality is quite good. Run TV audio out to the transmitter, the transmitter out to an analog AVR input.
All the best!