Shot placement trumps "flat shooting" every time as no-matter what cal you're thinking about, there's really no such thing as "flat shooting" once out beyond 300 yds or more. I'd suggest saving your money, take advantage of the 6.5 bullet choice and learn to home load to up the performance from your 6.5 x 55 which can take higher pressures than much of the factory ammo is loaded to. If it's good enough for those taking deer out past 500 yds across New Zaland's steep sided valleys where that's a pretty realistic range many are shot out there, it's more than good enough for the UK. Even the CM retains more than enough energy to bang flop a deer at 300 to 400 yds, but once out at those distances, the HILAR shot maybe the most reliable bullet placement and offers a large enough killing area to make moa accuracy enough for the job. I've shot into under 10 inches at 1000yds with a factory CM and no doubt might expect similar with other 6.5's using high BC bullets.
I'd be more concerned about bullet placement though and bullet selection and learning to shoot consistently accurately and not fretting about any paper advantages of one cal over another. That's become a fanciful topic on many forums when the truth is those shooting to 400 or even 500m on occasions are probably in field conditions might not be capable of 1moa accuracy on target consistently. You ought really to be more concerned on things like knowing your drops, effects of wind and being capable of determining your scope adjustments accurately irrespective of cal to put the bullet consistently into the right area. Personally, I wouldn't shoot deer atthose ranges when for the most part, in the UK, you ought to be able to stalk in closer, albeit hill shooting may on occasion only offer a 300 to 400 yd beastie. That demands accuracy.
The advantages of a 6.5 as the OP suggests is that it's a lovey forgiving cal to shoot in all flavours. The differences between them are real world marginal for hunting. Target shooters will gravitate towards the ever popular 6.5PRC because of its MV advantages with long for cal high BC target bullets offering improved ballistic performance but for hunting, I wonder what the factory ammo availability is compared with CM or 6.5 x55? Then there's the cost of re-barrelling. Push any 6.5 hot and you're going to be looking at more frequent re-barrelling than say a .308.
In your shoes? I'd keep what I have and home load to benefit from the advantages of the powder column and pressures you can load the x 55 to.