FFS: Eccleston made his decision on whether to stay or not halfway through the season shoot, before John Barrowman arrived on set. Anything which influenced his stay/no-stay decision happened during the shooting of his first half dozen episodes.
One clarification.
Eccleston didn't make his final decision until the eleventh hour -- shortly before "Rose" aired and as the production was gearing up for the second season. The man was genuinely torn about returning; he liked
being the Doctor, he didn't like the behind-the-scenes crap he had to deal with.
What finally forced him to decide that no, he wasn't going to return, was that he learned the BBC had put out feelers to Tennant about taking up the role midway through the second season. Eccleston felt he wasn't wanted, so he decided to leave. In the BBC's defense, they didn't know what Eccleston was planning on doing and they were only protecting their investment in the series by lining up another actor.
Part of the reason for the glowy regeneration effect was that it allowed the production team to mask Eccleston in the shot. The regeneration scene is actually all Tennant; they were able to match the original ending of "Parting of the Ways" to Tennant wearing a skullcap.
So, yes, the main events that soured Eccleston on working with the crew of
Doctor Who happened early in the production, but there was still a chance he could return -- until the BBC scuppered that.