Repair this shower tile problem

Situation:

I am trying to restore damage to a shower. Turns out that the upper-right corner (at orange arrow) had popped proud of the surface (some weeks or even months back), allowing water entry along the large breaks in the grout seal. No one said anything when that happened, so…

… here we are. There was also some previous damage to the corner tiles (white arrow), which had been "fixed" by previous owner with a slathering of clear silicone.

I removed the popped tile. The backer there was plain sheetrock, which disintegrated while removing the tile. Backer on the leftmost tile (left of the corner) was a bit damp, but is cement board, is drying and is in decent shape.

I removed the remaining debris, dried it out, and I first thought I'd just attach a piece of Durock cement board. I attached the Durock to the stud in the corner and the one a few inches to the right of it. However, the right edge of the Durock (white arrow) was unsupported, and flexed with the slightest finger pressure. A terrible idea.

I removed that Durock piece. Suspecting that I might have to go rightward to the next stud to attach the Durock at both left and right, I measured to the the right and found the next stud. Next image shows how far I'd have to go:

One Idea:

Remove the two tiles to the right to expose half of the next stud, then attach the Durock from the corner to that stud, giving a (reasonably) solid substrate to replace the (now many) tiles with mastic, then re-grout.

Not sure what else to consider. I can't attach a tile to nothing, and the unsupported Durock idea was terrible. If there was only some way to somehow support that right edge of the pictured Durock piece solidly enough to be a decent substrate…

Considerations

1. I can grout, caulk, and cut tiles with a manual tile cutter. I can cut or shape practically anything with a die grinder and/or diamond abrasives.
2. Low cost is important. Replacing framing, large areas of backer or tiles, or "Just retile the entire bathroom the right way" are not good suggestions.
3. I'd like to repair this myself, and soon.
4. I only need a reasonably good fix, not perfection. The original sheetrock-only construction was not exactly robust. I just need something that will offer a solid enough substrate that this repair will last some years and prevent further water intrusion.

Question:

1. Is the solution I mentioned above a good one, or are there far better ideas I haven't thought of? Some way to avoid removing more tiles and somehow support the smaller piece of Durock adequately to a reasonably solid substrate?