The nine-year-old, having initially dead-heated for first in the Stella Artois Novices' Handicap Chase on January 4, was placed second after stewards found he had interfered with Pinch Of Ginger on the run to the line.
Following a further look at the race, Oneida Tribe was judged to have crossed the line in first, but the result remained unaltered due to the original decision of the stewards.
Dickin said: "On professional advice I have decided not to appeal. The likelihood of me actually winning was probably slim.
"It's gone on and on now and everyone is bored of hearing about it. I don't think my opinion will change the stewards' opinion, therefore I am not appealing.
"There is no way my jockey was careless. In my opinion he never had the whip in the wrong hand any way at all.
"I've had a lot of people in the trade saying I ought to appeal and I have had a lot of people give me sound advice.
"If I took the case to London it would take up time which I don't have and if I lost it I would be more annoyed than I am.
"I want to put it behind me now. I've been given the race, had it taken away and I don't want any more of this. I will get on with tomorrow.
"I am not a stewards' room person and rarely do they have trouble with me. I don't have to agree with the decision and I never will.
"It was a bad experience and a bad decision, but it was their decision and I have to respect that. I will let the horse do the talking and as he stays so well, he will probably go back to Towcester over three miles on Valentines Day."
Adrian Heskin was in the saddle for the Bangor race and Dickin has not ruled out teaming up with the jockey again in the future.
He added: "I think he has earned a seat. We've got to know each other quite well over it and he has been here to ride out for me once.
"Charlie Poste has always been my jockey and Jack Quinlan rides a lot of my horses in work. Adrian and I have made a connection, he is a really good chap and a talented rider."
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