I don't think the .338 will do excessive meat damage, quite the opposite. I have a .375 for use in the UK and believe me a .300 grain soft hardly opens up and does little damage, I think bullet construction has more importance than calibre.
To answer the OP your rifle would be perfectly adequate to shoot red deer, however long range shooting is generally frowned upon in the UK. For some reason a person who has practised continuously at the range and has worked out the limit of his/her ability is unethical yet the people who never shoot more than 200yds are generally the ones that will shoot them in the guts. I have seen far more deer gut shot at close range than at 300/400 metres.
I own a lot of commercial forestry and rely upon it for an income and deer unfortunately are a pest and providing I am confident with the conditions (I.e. the wind not continuously gusting from 20-50mph) I will shoot deer at extended ranges. However when I go on my annual trip to stalk stags then I really enjoy the stalk and crawling really close.