If the leak is in the ground. No way, to repair it, without digging. I am sorry.

Could you snake a flexible pipe through existing pipe? But they probably used flexible pipe so probably won't work.

You said the pipe is leaking but if you haven't already tested the basin I would....fill the bottom basin with water, cover it (plywood or plastic?) Then check back later to see if the water level drops below the pipe. If it doesn't drop below the pipe then the issue is indeed the pipe. otherwise the leak could be in the tub/basin.

How fast is the leak? I'm not an expert but the design of this water feature looks like it would allow for a relatively large amount of evaporation (high surface area to volume of water).

Good advice from kstug. Wide shallow spillway, lots of paving around the basin, and judging from the hills in the background of the first pic you probably get a decent amount of wind. I have a similar water feature in my yard but I only have an 8" wide spillway, and I lose a decent amount of water through evaporation and wind spray.

The best way to troubleshoot these is always to eliminate possible causes. Do what kstug suggests and eliminate the basin as a possibility. Then run a flexline from a submersible pump into the rill to eliminate the plumbing as a possibility. At that point you'll have a better sense of where water's leaving the system. Getting a water feature repaired is hard and expensive because the repairing contractor is assuming responsibility for something they didn't build, and the only thing better than water at getting where you don't want it is a hyperactive kitten in a combination aquarium/yarn store.

It looks like something that is very cool and desirable in a blog post/Pinterest page/sales brochure/model home/HGTV show but turns out to be impractical and therefore hardly used in real life. Just like an old fad, big jacuzzi tubs in master baths (too much time and hot water to fill) or a current fad, the front yard outdoor patio/lounge area (who wants to sit in their front yard in full view of the neighborhood when you have a more private back yard?)

Appears to be a safety hazard--a good place to break an ankle. I would fill it in and forget it.

So what does it do? Does water come out of the bar area, travel down the middle of the bar and spill into the tray at the end? Very different.

So when the fountain is running, the water in the basin shown drops until it is below the level of the pipe, then the water just sits there?

I'd check the seal around the pipe.

I'm pretty sure I didn't say anything about it being in your front yard. My comments were more general, based on many people I've talked to who craved certain currently faddish features that looked cool but then rarely used them. It's great that you use it.

Errr, kentc.... the front yard outdoor patio/lounge area (who wants to sit in their
front yard in full view of the neighborhood when you have a more private
back yard?)