Generally, when an appraisal comes in lower than the contract price, the seller immediately splits the difference 50/50, and you can haggle from there. The fact that they only offered to go 15% down from the higher appraisal is a red flag. Seems to me that they invested too much into the property and now have an overpriced property for the area. That's their problem, not yours.
Have you had your inspection yet? Part of them wanting to keep the price higher could be the fear of inspection repairs/credits. What I would do if you really love the property is tell them you'd be happy to pay $400k with the property as-is (still do the inspection but if the costs are reasonable, you'd fix after purchase). This sounds like you're offering them something when you still have an out if the inspection items are too much.
As others mentioned, they will have the same problem with any buyer and they know it. Don't let them trick you into paying more than you should. I once had a house under contract where the appraisal came in lower and there were foundation repairs needed. I told them to drop the price to the appraised amount and fix the foundation. They said they couldn't afford to do both and gave me the line that "there are other buyers lined up", so I said I'd pass and they can go to the next person in line. Two weeks later they came crawling back and wanted to help "fight" the appraisal. I told them I had moved onto my next property already.
Best of luck!
