If you've fixed the visual damage, that's all well and good. From my experiences, the browning seems to have appeared a little early to be cellulosic or alkaline browning. Has the carpet got a jute (Hessian) backing?
What pH was the solution spilled in it by the way? Have you pH tested the affected area to make sure it's neutral?
Uplifting the carpet is probably the best way to go, to be honest. There may well be moisture lurking down there, between the carpet & underlay, in the underlay itself, under the underlay (stop me if I'm sounding like a Mexican) and even soaked in to the subfloor. Moisture + warmth = heaven for mould and bacteria, which are causing the smell.
Before relaying the carpet, make absolutely sure that the subfloor is dry (it should be treated with a microbiological inhibitor when first exposed). If this had been a longer-term escape of water such as a burst pipe or higher volumes of water you'd need dehumidifying equipment to dry the floor off, but seems as it was a low-volume localised spillage for a short time, you'd probably be fine just having an airmover on it overnight (yeah they'll love that, I know).
Replace the affected underlay - it's quicker and more practical than messing around trying to dry and sanitise it).
Refit the carpet which should have been stored so air can get to the affected area. Obviously if you're like me and wouldn't attempt to fit/refit carpet, you'll need your friendly local carpet fitter on side.
That should still work out cheaper than an insurance claim, if your excess is hundreds.
Only one thing though, sometimes excess water can delaminate a carpet, it might not be immediately obvious. You don't want to go through all that and then the customer rings you up a week later and says it's gone all puckered up.