Ethernet 7

If it is a true CAT 7 cable then it will have isolation/grounding between the pairs of wires.
You are going to need a female to female coupler that has a grounding feature.
This should be a good coupler:
https://www.amazon.com/AMPCOM-Coupler-Keystone-Shielded-Female/dp/B08PRFGS1P/ref=sxin_10_ac_d_pm?ac_...

But, I would contact INAKUSTIK first, to be sure.
Since he is adept at electrical configurations, I would PM member Jea48 and get his take on this.
My experience is limited to my recent foray into rewiring my ethernet cabling.
Bob

I just thought of a more direct way.
Why not splice them together?
https://www.bestethernetcables.com/how-to-splice-ethernet-cable/

The wires are color coded so it wouldn't be rocket science to perform.
You'd need the proper tools, though.
Though I would understand your not wanting to cannibalize your cables.😮
Bob

Just get 50 feet of Cat 6a. 

Well  you could buy a coupler like this:
https://amzn.to/3qn1l4d
but why not just buy 50' of Cat 6??

Better still stick a network switch in the middle rather than run a cable directly from router to streamer. The switch (ideally with quite power supply) will act as a buffer between router and streamer, blocking some of the noise from the router. The switch won’t block all the noise as it will emit some of its own but a lot better than not having one. :)