For wiring, you should not be looking at the projector as your audio switch and controller. It is just not what they are designed to do.

I would get a HDMI switcher with audio extractor and put it up front with the sound bar.

Something like this:
https://www.amazon.com/J-Tech-Digital-HDMI-Audio-Extractor/dp/B083WVFZBP/ref=sr_1_3

That will allow for up to 4K 18Gb/s HDMI connectivity and should give you a proper 5.1 audio output to the sound bar.

From the switcher, you run a single HDMI cable up to the projector.
I strongly recommend you put in conduit (1.25" or larger) between the equipment location and the projector so that when (not if) you need to upgrade the HDMI cable you can do so easily.

All this said, soundbars, while better than speakers built into projectors... well, they suck.

If you are doing a dedicated room AND you have the walls open pre construction, you should go ahead and wire for a 5.1 or 7.1 audio system. Maybe more. You don't have to buy anything yet, but the audio quality of good speakers makes a huge difference to the entire experience.

This is a conversation that can be had, and I'm not sure of your room setup/construction to know if that's all feasible, and certainly I approve of a soundbar over speakers in a projector by a mile. But, would rather see a good pair of floor standing speakers and a subwoofer over a soundbar any day of the week. Fake soundbar surround sound is just not very good.

Also, full surround systems can be had for just a few hundred bucks. Good in-wall cabling from Monoprice can be about $150.