I'm not a pro, but my first thought is it will take quite a bit of heat to melt solder on that thing! I'd be worried about too much heat, and you probably don't have a torch big enough.
I think if I was going to do it, I'd remove the copper and do as they indicated - use a donut and replace the stuff with pvc. If you cut the copper off near the cast iron hub, you could probably rip that joint out of there. The pros may have some suggestions. With cast iron, (I've only done a couple), I used a drill to make swiss cheese out of the lead, then pryed out the lead. With the copper, you could probably use a hack saw or sawsall to cut through the copper from the inside and some of the lead, and fairly easily remove the whole thing, especially if you made two cuts and pryed out a chunk. The lead will stick to the copper, but it will come right off the CI. Then, a wire brush to clean it off and maybe a little dish soap on the donut, and you press in the pvc stub, and glue up the elbow, and riser, and you're ready to install the flange. Get one with a SS ring, and you will need to drill some holes in the tile to anchor it to the subfloor. This may require a diamond bit. Lowes carries some. Get one the right size for the anchor you use, or if you can catch the plywood, ensure the hole is big enough so the screw doesn't catch or you'll crack a tile.
You could probably have the whole thing out of there in 1/2-hour, and, new stuff inserted in another 1/2-hour, depending on access and the skill of the operator. It'll probably take longer to drill holes through the tile to anchor the flange.
Again, I'm not a pro, but my guess is it shouldn't take more than 2-hours labor, so at $1000, unless they're doing other things or you live way out in the boonies, seems a little high. Could be normal, different locales have different rates to account for different cost of living.
Parts to do this are probably less than $25 - less than you paid for the copper flange. And, you could recycle the copper you take out. It's proably at least a couple $/pound.