I am always happy to share. I have no trade "secrets".
But you already guessed 90% of it Englishman.

That's almost exactly how I do it. But I do not use wire however. That won't work. The wire is too weak when it get hot to hold tight.

Here is how it's done:

Cut your box to what ever shape you want. Perch belly or flat. Next take a piece of 1/16" steel sheet metal and cut a plate out that will over-hang the box by at least 1/8" all around. Take pains to get the bend just right so you can see almost no light between it and the box. Gaps should be nearly non-existent. Sometimes this alone takes me about 30-40 minutes to get a tight fit.

Now, trace your box on a piece of hard maple wood, and cut it out carefully to fit the contour. Wood shapes fast with a file and scraper, so it only takes about 10 minutes to get the wood to metal fit exactly.
The block needs to be wide enough to saw it in 2 pieces so you will have a "clamp" for each side of the magazine. You will braze one side and then the other. When your floor plate and your wood blocks both match the box bottom perfectly soak the wood in water with some dish soap for about 1 hour. You want it very wet.

Now you are ready to make the mag.

Clamp the new floor plate to the box with the wood, holding it sideways in a metal vice so only one side of the box is above the vice jaws about 1/4". Use only light pressure. It need not be tight as long as you make things fit well.

You will need to work with a 0 or 00 tip on a Oxy-Acetylene torch. Take 1/16" brazing rod and heat it a bit so it will "grab" some flux, about 2" up the rod.

Start with the center of the box and plate and heat the bottom of the plate first to a red heat and then the box wall. When you have about 3/4" of it heated apply some rod. The rod will flow sharply into the joint and the wet wood smokes and burns, but not fast enough to give way. This all goes pretty fast. It takes me about 45 seconds to 1 minute to join the plate to the box from center of the side to the center of the back, and also from the center to the center of the front. About 1 minute TOTAL time. Work one way and then the other, until you have 1/2 the box brazed. Work from the middle out. That keeps the metal from warping because as it cools and shrinks it pull inward, instead of pushing to one side as it would if you start at one end.

When you pull the flame back the brass will cool to a point it is solid in only about 5-8 seconds. Take the mag out of the vice and let the scorched burned block fall. It's used up.
That's why you made 2 at first. When the box is no longer red you can quench it in water.

Now turn it around and do the whole process again with the new piece of wood on the other side.

If you pull your flame back from the edge after you make the joint to the bottom of the new floor plate the brass will follow the heat inside the mag and make a water tight joint.

To test, set the mag in a pan of water with dish soap for about 5 minutes. If the inside stays dead dry your joint is perfect. It's also VERY strong.

Now cut the "flange" of metal off, to give a flush fit on the box, and file/stone/sand off the brass. Give a slight bevel to the edges of the plate so it comes to about .040" at it's edge instead of the .062" you started with. It will look just like the ones that came from London. Polish it up and you are ready for engraving.