Let me extend my sympathies to you. I think a lot of folks hang onto a fantasy that as your folks get toward the end of their lives that you'll somehow be more ready to lose them. We're not too far off age-wise. I'll never be ready to attend my mom or dad's funeral.
That wasn't your question, and I don't have any direct experience, yet (both of my folks are in relatively good health at the moment), but I have some thoughts if it's worth anything.
Our Dads sound like they'd get along. Were this my dad, I'd save my breath before trying to convince him that he's too old to continue with the stairs. My Dad required back surgery about 15 years ago -- almost a year's recovery and was the worst medical experience of his life. He spent much of it barely able to move (complaining so little that you'd have no idea most of the time). You don't ever really "go back to normal" after surgery like that. You also can't stop my Dad from pushing someone out of the way to get on one end of a couch being moved, or keep him from bringing over his tools when he finds out one of his kids is working on their home. It's a losing battle. And I've learned that while I may believe he's doing more damage (and he probably is), at his age, he knows his limits and will exercise them. One way to guarantee he sets those limits too high is to hint that maybe his body can't handle it.
The bedroom thing is interesting, though. I'm assuming your parents mental health isn't declining enough that you suspect the stubbornness is caused by dimentia. If that's not the case, are there other reasons that you haven't shared (or maybe you haven't thought about?) Is there something of sentamental or emotional value about that room that keeps him wanting to remain upstairs or is it just conveinece? Either way, the incredible inconvenience of having to crawl up and down stairs has apparently been judged "worth it" to him.
And I get the unwillingness to move. Even at my age, I've my elderly grandparents/great uncles/aunts and some other extended family have moved from "the family home" to "the home they're going to die in" (usually something smaller, sometimes assisted living). Most passed away within two or three years of that move. Even though I can rationalize that "of course they did -- many of them were moving because they were too frail to remain where they were", when you've watched the progression from "home->condo->grave" happen so quickly for so many, it's hard not to ignore the other variables. If your Dad has lived in this home for a lot of his life, it's the home he's "lived in" -- the next one is the home he'll "die in". I believe it would be impossible for me to make that move without feeling like the only thing you're saving up for anymore is a funeral you won't be conscious to attend.
Unlike you and I (hopefully, anyway), however, your Mom and Dad have probably not gone very many months without a person they know/knew/grew up with ending up on the obituary page. You're literally confronted by your own mortality all around you -- moving to "that last house" makes it part of the rest of your life in a way that accepting "I'll die where I lived" does not.
... when I was younger, I always wondered why people go through the expense of installing stair-climbers in their homes -- the inconvenience/grief of strapping yourself into a chair and having a machine take you up/down because you're too frail to manage the stairway and the expense of having one of those installed seemed crazy compared to the benefits of just selling the place and moving somewhere without stairs. As a younger guy, I would always weigh the cost of upgrading my home against the cost of moving somewhere that already met my needs -- especially if moving would result in me making money (as such a move from owning a larger to a smaller house in about the same location would cause).
So twenty or so paragraphs to summarize with this: Recognize that your father's motivation is almost certainly a combination of: fears about his own mortality, unwillingness to move to "the death house", unwillingness to accept that at some point people will need to take care of him/he will not be able to take care of himself, unwillingness to leave the memories of his life behind from within those walls. If you can help him to understand that moving/correcting some of these issues means having more of that life to enjoy with his family and loved ones, you might be able to convince him to choose circumstances that are easier for him to live with.
Either way, enjoy the time you have with your folks for as long as you can.
[0] Well, those who came from stable family upbringings and have strong relationships with their folks; obviously there's many exceptions, there.