So, without getting all that into it, the reality is that I've never used any HDMI extenders w/ARC or eARC functionality at all. I've very specifically avoided them as they rarely make a lot of sense in my installations. ARC is often buggy and headache inducing and HDMI-CEC rarely works in a manner that is fully useful.
That doesn't mean I haven't done a bit of digging, and products like this should actually work alright for this type of purpose...
Buy a HDBaseT 4K HDMI Extender upto 132 ft with ARC Support from OREI with a 30 day return policy, 1 year warranty & free shipping in the US.
While I haven't used this specific product, I've used gear from these guys before and it has worked as expected. There are similar products out there as well, but at a price of about $260 or so, this is right about where I would expect it to be. Right now I think HDBT 3.0 gear which might work is significantly overpriced and doesn't seem as directed towards the consumer market space.
Since a cable which supports 4K would be necessary as it is, I would expect that this may have been necessary at some point as it is.
I would also TEST with the existing HDMI cable as HDMI ARC is not a feature which takes up significant bandwidth and may be supported by existing cabling without the need to buy anything at all.
At the end of the day, I would also take a hard look at the AV receiver as that will need to get onto the 4K bandwagon at some point.
If wanting to hit a better price point, you could ditch the ARC audio and instead use the optical audio output of the projector and just use optical audio baluns to just send audio back to the AVR.
Buy a Digital Audio Over CAT5 Extender Upto 1000 Feet from OREI with a 30 day return policy, 1 year warranty & free shipping in the US.
That'll save $200 and I expect may be less of a headache than trying to get HDMI ARC to work perfectly.
A shame about the projector choice... Lots of Optoma issues as of late. Especially their UHD series of 4K models.