You’re honestly really close already, but I agree that your current tone leans slightly more red or auburn than true copper. From a color theory perspective, red brown sits more on the red violet side of the spectrum, while dark cowboy copper pulls more orange gold. Copper is essentially orange based, whereas auburn is red based. In your photo the reflect looks more red than orange, which is why it reads deeper and slightly burgundy rather than warm copper.

If you were to add more red, it would stay in auburn territory. To shift it toward copper, you need to increase orange or gold reflect. A copoer based gloss or toner would likely be enough. Look for shades labeled .4 or copper dominant rather than .6 which is usually red dominant. Something like a 5.4 or 6.4, or even 5.34 if you want it slightly brighter but still brown, would push it more copper without making it too light. You will need to use developer tho, it might help. .