Pompadour is a romantic, period-correct pink — and like most historical pinks it's at its best when you stop it going saccharine. The trick is contrast and restraint.
Start with a chalky off-white to give it room to breathe. Paint & Paper Library Sand I (LRV 95.4) is about as airy as it gets — a near-white that keeps things light without throwing a cool blue cast back at the pink. Farrow & Ball All White (LRV 92) does a similar job if you want something cleaner and less creamy. Either works on ceilings, cornicing and woodwork to frame Pompadour rather than fight it.
For drama, go deep. Pompadour loves a Prussian or inky blue as a feature wall or in joinery — that classic pink-and-navy pairing never dates. And for warmth and grounding, Mylands Cigar BH.20 (LRV 11.8) is a gorgeous near-black brown that gives the scheme weight; think skirtings, a fireplace surround or a single bold cabinet.
If you want a metallic-adjacent accent without literal gilt, Dulux Copper Glow (LRV 30.1) brings a soft burnished tone that nods to antique metals. Use it sparingly — a small piece of furniture or a panelled detail.
The "but what about brass?" question comes up constantly with pinks like this. Gilt is absolutely acceptable here — but only antique, aged or burnished gilt. Bright modern polished brass looks brash against Pompadour and pulls the whole room toward bling. Cool metals (aged nickel, pewter, soft chrome) actually flatter it better than warm ones.
Practically: pale, unfussy woods — limed oak, ash — sit far more happily with Pompadour than dark mahogany, which can muddy it. Test your off-white and pink together on a board before committing, because north light will cool the pink and evening warmth will deepen it. Two coats, proper drying time between, and let the white do the heavy lifting on the trim.