Ok, this past weekend I replaced the hard brake line.
I have an 03 base Vibe with ABS (That may have been a standard feature but I see places in service manual where they have non-ABS diagrams).
I ordered the 03 Matrix part # 47315-02180 which is "Tube, Front Brake, No. 5". (About $9 plus shipping). I cannot find the diagram on the web that allowed me to figure out which tube I needed, but it was similar to the one below. Tube #5 is not labeled, it attaches to what is labeled with 8 in a circle. (source: parts.toyota.com)
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The actual replacement was not too difficult, just messy. As suggested, I got the cruise control mechanism out of the way. I did not undo the cables, I just removed the electrical connector, unclipped the vacuum tube on top, and undid the two bolts securing the base to the mount on the car. These bolts were fairly tight and rusted and I couldn't really get at them with things being in the way. I ended up using some WD40, an extension with a universal joint and a 12mm socket. Once I got the bolts off, I had to lift up and jiggle the assembly a bit to free the tab on the base that fits into the mount. There is a wire standoff that holds the throttle cable looping in on the bottom, which I unclipped, and rested the whole thing to the side on top of the engine cover.
Now I had access to the fitting I needed to undo. I put a rag underneath to catch drips. They say to leave the brake fluid reservoir cap on to provide some vacuum so the fluid doesn't come out, but it started dripping nonetheless as soon as I unscrewed a few turns. It's a bit of a pain, can't easily turn it with finger strength but i finally got the fitting off and had my assistant (wife) plug the hole with her gloved finger while I worked on the rest.
I undid the brake hose at the hard line in the wheel well and pushed the rubber plug through towards the engine bay to free the old part. Putting the new part on was harder than taking the old one off, had a hard time getting the new fitting properly threaded on with time pressure while the fluid was coming out. Eventually I managed. (i attached the new hose first but I had to undo that because there was not enough freedom of movement in the hard line to align the fitting). Anyway, once the hard line fitting was threaded on in the engine bay, I put the new hose on, new copper washers on the banjo bolt, bled the brake line, topped up the fluid, etc. Putting the cruise control back was exactly the reverse of taking it off and easy. The most annoying thing in all of this was the leaking brake fluid - I think I caught most with the rag, but I wiped down the surfaces as best I could with a paper towel and then decided to pour some water over everything anyway, catching it in a drain pan. (For completeness, pushing the new rubber plug that is on the hard line into place was also annoying. I had to both push from the engine bay with one hand and pull gently with pliers from the wheel well with the other hand, basically hugging the fender and barely seeing what I was doing.)
That's it. I don't know if this will solve my original issue, I'll be testing it, but at least I now have a new brake hose and a hard line with good threads on the fittings.