This may not be an acceptable answer, but I don't believe that you are going to be able to find a BIFL couch with a budget of $2k. That is a lovely aspirational goal, but it is not realistic.

If you truly want BIFL furniture, you need to look at what is known as "contract quality" furniture. It is often sold only to the trade, but you can also find it at certain retailers. Design Within Reach is one source, if that matches your aesthetic, which is likely since you previously shopped at Ikea.

For traditionally styled furniture, look at the North Carolina furniture mart, which supplies a lot of the hotels. It is a mall-style outlet which I think has an online presence now, so many suppliers are together in one place. There was one vendor with contemporary furniture among the hundred or so, the rest were all traditional.

These are likely to be out of your budget range. If you truly want to BIFL, then upholstery is not really that difficult to DIY. If you can pick up a vintage piece with a solid hardwood frame, then you just need a few simple tools and some time spent on youtube to strip it down and redo it. The supplies are readily available thanks to the internet and you get to choose your upholstery. [Again, go for contract quality! Do not skimp on the fabric!] www.foambymail.com is a great source for quality foam.

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Good luck!