Another couple of options for you.
If the barrel is 21", and you don't mind it a little shorter, you can have the barrel cut behind the muzzle break, recrowned, and threaded. No heat required, and might not have to be refinished.
Second option is replacing the barrel. For my FAL, I ordered a NIW Imbel Paratrooper (18") barrel and am having that installed. Chrome lined bore, chamber, and gas tube with a 9/16-24 LH thread already on the end. Runs about $180 for the barrel; installation costs vary. If you go this route, make sure you are not busting the foreign parts count. DSA will be able to tell you how many foreign/US parts are in your rifle from the serial number.
Sell the old barrel on the EE or keep it as a spare.
(or send it to me and I'll watch it for you) Herewww.falfiles.com/