Having shot a considerable amount of both Red and Sika stags with clients using all sorts of calibres on both Scottish and European Red Stags, some of which have been in or around 400lbs, I can honestly say that I have noticed no great overall difference between using a 243 through to a 338 calibre PROVIDING the shot is placed correctly.
A great many of my clients are American, and some prefer to use two of my rifles which are a Ruger Mod77 stainless in 25.06 and a 270 Tikka, using 120gr and 130 to 150gr nosler/sierra bullet respectively. I have just changed to a new cammo Tikka Lite in 270, good working rifles IMHO very pleased with it.
Again the subject of calibre has been raised. BUT at the end of the day it is all about correct shot placement, and knowing your quarry. You can badly shoot any species with any calibre (and belive me I have had clients that have) and the deer will still run like the wind. I knowpeople who have used a 243 in Africa on Zebra, and killed them dead!! Again shot placement, although I would not recommend a 243 for this job.
The 243 is an excellent round, but to answer the question asked about Red Deer. If a client of mine was booked to take Scottish or European Reds with myself I would advise him/her to bring a 25.06, 270 or 30.06 or any comparable calibre, you really do not need anything bigger to achieve the desired results. IMHO the 25.06 is a very under rated calibre in the UK, being a necked down 30.06 case. I have taken many African species with this calibre including Wildebeest, which are renowned for being tough.
I have had a number of American clients with 300Win Mags. And have had Jap Sika Stags shot wiht this calibre still run off, even though shot placement was correct (that is Sika for you).
I personally do not rate either the 25.06 or 270 as under gunned for Red Stag, having used these two calibres for about 20 years professionally guiding clients. Red Stag are not to my mind that tough, not when you put them up against a Japanese Sika Stag, which pound for pound will take a phenomenal amount of punishment sometimes. From my memory over the last 15 years (my memory is not waht it used to be when I try a remember back any further) I have had both species hunted with the following calibres with clients. 243, 25.06, 270, 270weatherby, 280, 30.06, 303, 3030, 300win mag, 338. Plus a few European calibres, as 90% of my clients are Americans.
Shot placement gentlemen and knowing your quarry, along with terrain and weather, these are the important points to remember.