HDMI Extender over Ethernet recommendations

Yea, I would be happy to give up on the Fire TV if that proves to be an issue.   Some of the cheaper 1080P/4K HDMI extenders I've been looking at mention the Fire TV may require the higher model 4K extenders.

I wish I had used tubing through my room's cable path to let me pull another cable easily through the ceiling; I would then try a newer HDMI cable before anything else.   Oh well, live and learn...  I suppose some day the Cat6 cables will need to be replaced too :(

I just purchased a refurbished Key Digital KD-X222PO Video Extender to try with the Cat6; I can return it if it doesn't work with my setup.

Its specs look very good:
› Signal Extension:
» Up to 125 ft. @ 4K 24/25/30(4:4:4)/60(4:2:0) using third-party CAT5e/6 UTP/STP cabling
» Up to 230 ft. @ 1080p / 1920x1200 using third-party CAT5e/6 cabling
› HDCP 2.2: Compliancy up to HDCP 2.2 and backward compliant
› HDR (High Dynamic Range): More life-like images through a greater range of luminance levels
› Deep Color Support: Up to UHD/4K 30Hz 4:2:2/12 bits or 60Hz 4:2:0/8 bit 
› HDMI® and HDCP Licensing: Fully licensed and compatible with HDCP 2.2 and HDMI latest technology such as 4K/UHD 4:2:0/8bit at 60f/s

Hopefully it will just work!!

I'm still holding on to the Emotiva processor just in case.

Re-reading some of your posts, I have some additional comments. 

I have found that Amazon's newer O/S for Fire TV may not necessarily work with older processors/receivers.  I had some problems getting my older FireTV unit to work through my Krell S1200U-3D (which is an HDMI 1.4a hardware).  You may want to try another type of streamer such as Roku, or invest in a better Zappiti or Zidoo streamer.

The new Yamaha receiver may be trying to do a 4K HDMI check.  That Beldon Series 1 cable is really not that great of a cable.  True, it is 23.5 awg for conductors and 24awg for control wires, but the copper is pretty poor quality (not even OFC), and the control wires use PVC insulation which really absorbs electrical energy.  It is likely that the Yamaha's HDMI signals are just not getting to the projector properly.

HDMI cables are  tricky bastards.  I once did a long run and it was an effing nightmare.
I have a drop ceiling in my basement and it’s perfect for hiding the Ethernet, speaker, and whatever wires for the basement system and the living room system right above it.  My 2 channel system in the upstairs has the shortest cable runs possible without going to an integrated amp/DAC.
  Do you still have the old Emotiva?  Maybe you can reinstall it while you try to figure it out

I actually have an HDMI cable strung there now, but it stopped working after I changed processors, and I cannot easily replace it.

I built my home theater room 7+ years ago and at the time ran an HDMI cable (BJC Belden Series-1 Bonded-Pair HDMI Cable)  from BlueJeans cables that was rated one of the best for long distance.  The installer I purchased my projector from suggested I also run Cat6 because he doubted the long HDMI cable would work.  We plastered the ceiling instead of doing a drop-down.    The HDMI cable worked fine with my Emotiva Processor and all was good.

I then got a newer source recently (Amazon 4K FireStick) and found it didn't work with my older Emotiva to Epson Projector (9700UB) video path.  So figuring an update might be in order I purchased a refurbished Yamaha TSR-7810 receiver which supports the newer standards and has a Preamp out to feed my Anthem amplifier.

After hooking up the newer Yamaha I found out none of my older 1080P sources (e.g. Tivo Mini, 1080P BluRay player) as well as the 4K capable Amazon FireStick work with the HDMI connection to the projector.  On every input setting the projector output is black with a flash of whatever should be shown maybe every 20s or so.

I tried the Yamaha output with an HDMI monitor (3 foot connection) and it works fine.  So I'm hoping it is the Yamaha's ability to drive the long HDMI cable vs the HDMI cable's 1.3a spec which I assume should work with the Yamaha.

If anyone has any other ideas about what I could try without spending more $$ please feel free to share.

It's blowing your budget, but you can also look at the Ruipro fiber HDMI cables.  A 12 meter (39.37 feet) is $149.

https://www.amazon.com/RUIPRO-4K60HZ-HDMI2-0b-Supports-HDCP2-2/dp/B0759CXK4H?th=1

Not sure how they compare to ethernet extenders, but the Ruipro has gotten very good feedback and excellent 4K HDR support.

https://metrahometheater.com/av-accessories-and-hardware/av-distribution/hdmi-extenders.html?limit=all

If you want true reliability and something that really works without headaches, Metra is the way to go.  You’ll spend more money, but you will have absolutely no regrets when using their products.