There is almost no circumstance I've been in where you can use an existing cable to pull a new cable. But, in this case, there is no need to.
FIRST: Why can't you just buy some wireless gizmo and be done with it?
DirecTV broadcasts professionally encoded h.264 or h.265 signals which have extremely high compression applied to them. Their 2160 is a long way from high quality 4K, but their encoders are top notch and add minimal delay.
When it gets to the DirecTV receiver, then it must be decoded. Uncompressed 2160p/60 can run in excess of 10Gbs. A typical home WIRED network often runs at 1Gbs today. So, that's 10 times the capacity of your current wired home Ethernet setup.
Clearly, getting a uncompressed wireless signal across the room, while not impossible, is far more difficult, and far more technical than you may have imagined. Now, if you put a 18" dish at your A/V receiver and a 18" dish up by your projector, I'm sure it would be a lot easier. You would have lag, but that's part of wireless!
SECOND: You don't need it.
There is no such thing as a 1.3b HDMI cable. There are only high speed and standard speed HDMI cables, and sometimes those ratings aren't accurate. So, the very first thing you do is you get some 4K content, and a 4K projector, and you try your existing cables. They may very well work just fine.
If they do NOT work find, then you move on to HD-Base-T (HDBT), which uses the cat-5e wires in place to provide a hard wired custom solution. They aren't cheap, but they are an industry standard, and the newest versions of HDBT can support 4K video. I use them throughout my home for 1080p without issue. The best part is that the cat-5e gets cut exactly to the length you need, so it is truly a custom solution.
This unit, by example, says it can deliver 4K w/HDCP2.2 support for $110.00
Seems like a small price considering the high reliability of a wired signal. Your original installer did a good job pulling the extra cat-5e cables.