Hi and welcome to the forum. ATC delays/restrictions can be a valid 'exceptional circumstances' defence, but it is all situation dependent. All of the other delay reasons you mention (airport handling, rotational delay and late positioning of cargo) are non-exceptional and BA should be taking responsibility for these.

One question might be whether the ATC delays would have occurred if the flight had been ready to push back on schedule. For example, if the weather got considerably worse whilst the plane was delayed on the ground, it could be argued that the ATC delays are a secondary result of the non-exceptional delays - and therefore also non-exceptional. On the other hand, if the weather was poor all day with long slot delays, that may well be exceptional circumstances and thus a get-out-of-jail-free card for BA.

I think you'll need to share a bit more detail of what BA have used in terms of their defence though, for anyone to be able to help you. There isn't anything personally identifying in the block times/route planning etc. documents so if they'll small enough, I think it's fine to attach them here. You could also attach the response letter with any personal details (names/ref numbers etc.) blanked out.

When it comes to these sorts of cases, BA does tend to throw the proverbial at the wall and see what sticks, so don't be overwhelmed by the mountain of arguments and evidence (purportedly in their favour) that they'll throw at you.