Bryan,
a couple of things to add to what has already been said.
Firstly, I don't think you need to stress on the re stretch thing because the main body of carpet is not being lifted, just the corner.
Harry may disagree with me, but I'd probably use either my knee kicker to re locate or else my mini stretcher.
Having said that, leave the overlap slightly long so you can trim to the right amount after you have stretched into place.
Another thing with it up against tiles, don't cut it too short or it won't hold in place. Instead of cutting with enough length to just go in with the edge in the bottom of the gully, cut it slightly longer and create a U shape with the offcut and you'll find that the 'doube thickness' while slightly harder to get in there, will tend to wedge it all tight and hold it.
Next, sell them on a clean of the area prior to doing your repair. It'll often give you a true colour to match to. After you have the donor piece in, use a little white chalk to lighten, you may even need a light grey crayon to dull the colour back a little bit, but that's more Tony's area of expertise.
Hope this helps.
John