Perhaps a sagey gray green? Is it truly purple or more of a cordovan color (which is how it reads on my monitor)? If it's a cordovan, one could do a lot of things to make it work if you like the color. I have a F&B color called "Pelt" in my guest room. I used white woodwork, white bed spreads, and yellow/white plaid taffeta for festoon blinds, dust ruffles and headboards. It looks fabulous.

I think the colors currently on the walls work fine. I might suggest however that you remove all the small understated wall and sparsely spaced pieces replacing larger more intentional ones for impact. If you opt to keep the art and pictures on the walls they need to come down to eye level height. Rather than painting entire walls you could paint the transitional walls like a bridge (the one by the stairs and the one by the front door). You could go with a latte, coral, grey, or gold. Good luck.

Sounds like a fireplace re-do might be a longer-term fix. In the meantime, I'd paint the mantle a dark brown, a similar colour to your end tables. It's not trim and shouldn't be white. It pops out too much in the white colour. Another idea that doesn't involve any ripping-out is to go right over the existing mantle using a mantle slipcover in wood. Here's one of many how-to's on the internet. If you stained your brick as well, you could have a whole new look. https://www.fwmadebycarli.com/2011/01/making-of-slip-covered-mantel.html

Do all the furniture pieces and carpet stay in the room? If so, take a look at Bittersweet Stem by Sherwin Williams. What don't you like about the colors you sampled?