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looking at thetrainline.com, you can get an earlier train from blackfriars, with a change in brighton that'll get you to chichester by 7am on a weekday.
an alternative to the train (which looks messy on the sunday morning) would be using the coach - national express runs from gatwick airport to chichester. so you could potentially get to gatwick airport, stay at one of the hotels at the north terminal (they're around £70 usually so that's down to your budget), and use the 0555 service 206 to get you to chichester.
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