I’m going to slightly disagree with the above comments here, based on my own experience of doing exactly what europeanbob is proposing.
My current commuter is an old, but decent MTB (653, full XT, old 8speed). I’ve been able to run road shifters successfully because 8spd Shimano road and MTB derailleurs are compatible. If this Carerra bike is 9spd upwards, then you’ll need to fit a road rear derailleur.
In terms of cost, I got a pair of 8spd Sora road levers for £21 on eBay. The drops were about £12, bar tape £7 and then a set of Schwalbe Marathon 26″ slicks were about £25 each. I also got some mudguards which were about £25.
I partly agree that a 2nd hand CX bike might be better, assuming you’ve got the money and space for a second bike. But don’t forget, for commuting you’ll need good puncture resistant tyres, and it’s unlikely you’ll find these on a CX bike. Same with mud-guards. So you might need to factor these into the purchase either way.
As for bar height, it totally depends on the size of MTB frame, specifically the distance between top of fork steerer and the ground. A largish MTB (e.g. 20″) won’t be much different to a medium CX bike. I suggest you measure this now, as if your MTB frame is small, you’ll end up needing a really daft looking stem to raise the height of bars. Looks do matter 😉
In terms of feel, my MTB doesn’t feel awkward at all with slicks and drops, but then it has rigid front forks (not sure if this makes a difference). It feels more stable going through big potholes/puddles compared to my CX bike (as you’d expect with 2″ slicks) hence the reason I adapted it.