GaryS

I anneal my own wire and the barbecue works well. I use regular charcoal and it's good to do after you've cooked a meal and the coals are grey and cooling down a little but are still glowing red.

I take the wire off the spool (I bought one spool of #6 grounding wire) or usually just buy it by the foot, like 30 feet of #10,12,14 wire and strip the insulation off.

Then I take a small coffee can and wrap the wire around the can carefully so it doesn't get all tangled up and twisted. It springs off the can when your through and then you can take some small #20 wire and tie the roll on both ends with that so it stays together.

Put it into the hot coals and let it glow red and let the fire burn down.

the wire will have a coat of black, flakey oxidized material on it and you can brush or shake it off.
What's left is soft dull red or magenta wire that is ready to use when it cools.

For that 3/4" branch I would some raffia and #6 copper wire to start and then you may need another length of #6 or maybe #10 to get it to hold.

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GaryS