There's a bigger question around the US-UK trade deal than the single issue of chlorinated chicken, or the associated issues like allowing GM food or animal imports that have involved growth promoters (growth hormones injected to artifically make the animal bigger).
The real issue is the right of a British government to regulate on what it does and doesn't allow in to the country. And even more so, the right of a British government to require labelling so consumers can decide what they want to eat.
The problems if the Americans push for similar things to what they were pushing for in the TTIP deal with the EU are that they wanted to lock down areas where they remove the right for a sovereign government to regulate. A democratically elected British government might want to ban a certain ingredient in order to protect public health, but then if that could be seen to threaten the profits of a US firm, the firm could sue the British taxpayer for compensation in a supranational court. That then acts as a constraint on the British government regulating in the first place - because of the risk of litigation, which is part of the idea, it keeps government out of regulating, which is why this type of trade deal has support from people that generally believe in governments staying out of markets.
Where it gets really sinister is in the US approach to labelling. A lot of free marketeers would take the position that it's not for government to tell us what we can and can't eat, so allow in chlorinated chicken, GM foods and so on but make them clearly labelled so then the consumer can choose what they want, and if consumers don't want to eat those types of things then the ones that can be signalled as chlorine free, GM free etc will sell more and the market will naturally reflect the consumers' preferences. Or you might get chlorine chicken selling at a lower price and non-chlorine chicken selling at a higher price because consumers will pay a premium. Sounds like a well-functioning market. But the US trade negotiators push very hard to limit labelling because they argue it could be 'discriminatory' against US imports - ie they want to remove the right for farmers that produce to higher standards to be able to distinguish their produce.
This isn't about free trade or making markets work better, it's about skewing the rules in favour of large US agrifood companies who will then be able to dominate the market, and preventing any action against them, either in the market from smaller farmers that want to compete on higher standards, or from governments bringing in any rules on public health/animal welfare/environmental grounds that might threaten their profits.
Now I have no problem in the idea that a democratically elected government wants to set the rules in its way, and then is accountable for them, and can be voted out of office by a new government that will change the rules. But in this type of trade deal, the UK government would be surrendering a lot of sovereignty and transferring powers away from a British parliament in terms of the right to regulate, to the jurisdiction of a supranational court. The irony is Liam Fox sits on Question Time constantly announcing that the British people voted for Brexit to take back control of our own laws, and get us out of the jurisdiction of the European Court of Justice, and now he's off talking to the Americans about this type of arrangement.