You haven't said exactly what resolution and refresh rates your displays are. I'm assuming HDMI 2.1 and 4k 120Hz but maybe they are older. The exact spec will determine what you may want to try in the end.
For instance, that first one is only good for 4k 30Hz and I would avoid it like the plague for that reason.
The second one looks a bit more promising as an all in one but is limited to HDMI 2.0 and at best 4k60 4:2:0. If that's all you need, it might be fine.
If you are actually looking to run 4K 120Hz, on both the TV and the projector, then I would say something more like this:
(The splitter I am currently using I believe and the only one I was to get to put out an actual 4K120 EDID compatible with my projector at the time)
And this off the HDMI going to the TV to extract the audio:
But again, this chain is getting kind of messy and it's difficult to predict if there will be odd interactions that prevent it from working correctly without trying it. If you're not running bleeding edge HDMI 2.1 rates, your options (and the chance to avoid complications) are likely to go up.
If you are dead set on using what you already have with no major changes though, something along this line is what you'll have to try without adding more serious pieces of equipment (like a receiver or processor of some kind - but even then I suspect you may need the splitter anyways).