Growing aviation is obviously not compatible with climate targets, but that's not what Heathrow expansion is about. A lot of the individuals claiming concern for this are just using a worthy cause as convenient air cover for their real passion: NIMBYism - it has a "bootleggers and baptists" feel to it.
The fact is, London has plenty of space capacity for origin/destination traffic across its many airports, where more flights could be added without a single word being said about carbon emissions. Just look at how Emirates and Qatar have ramped up operations at Stansted and Gatwick.
Expanding Heathrow is not about increasing emissions per se, but about making the country more attractive to connecting traffic, that is to say being the country that accommodates the global increase in demand, benefiting UK Airlines, UK retail from bored tourists in the duty free shops, UK hotels from overnight stops, and the UK generally from passengers who decide to stop over for a day or two.
Carbon emissions are important, but taking futile action is even worse. The planet is a global issue, and aviation is a global industry, so it makes no sense for the UK to worry about climate targets in the context of its role in global aviation (originating aviation is another matter, and that's why we have APD). We might feel better about ourselves for refusing to participate in global aviation growth, but the market would ensure that any reduction we do achieve would simply be compensated for by a bigger increase elsewhere - either in EU hubs, or increasingly the Middle East.