PETER NOONE has been booked for a starring role in pantomime this Christmas. He will play the title role in "Dick Whittington" at Bristol Hippodrome, which Tom Arnold and Bernard Delfont are presenting for 10 1/2 weeks from December 24. The Hermits will not be in the show. It is the same production that was staged at the London Palladium two years ago, with Peter taking over the part created by Tommy Steele.     While Peter is occupied in this role, it is anticipated that the Hermits will undertake live dates in their own right. Meanwhile, Herman and the Hermits together are set for a week in cabaret at Liverpool Wooky Hollow from September 19. And one week later, they fly to Berlin to guest in the TV special which Lulu is making there for British and German transmission, and for which the Carpenters were also set.     Peter was this week recording his single follow-up to "Oh You Pretty Thing" which will be released in October. At about the same time, the Hermits will have their first disc issued without Herman - already recorded, it is a Jeff Smith composition originally written with the title of "She's A Lady," although this will now be changed in view of Tom Jones' recent single bearing the same title.