Without actually realising it,
@caberslash has inadvertently given some good advice.
Kind of.
What I mean is this. The .308 is an infinitely flexible cartridge, with a vast selection of bullet weights and types. It is dead easy to download to modest velocity; it’s far far easier to do this with a .308 that with a .243 or 6.5 because you can use bullets specifically designed for lower velocity, like the 30/30 bullets. In typical UK roe hunting environments, there’s no ballistic penalty to speak off, with the correct zero range you’re still dead on at well over 200m.
It is something to think about, after all I do exactly this myself. I have several loads for the .308 ranging from subsonic to a super-fast 125gr, a heavy and slow 180gr, and a general purpose 165gr. A bullet that would be fantastic for small roe deer is the Speer Gold Dot at moderate velocity, I’ve shot small yearling fallow does with this at 2,600fps MV and it mushrooms perfectly with next to no bloodshot, knocks them flat as a pancake. Another great option for roe is to follow my cuz’s method and use the 125gr Sierra FN 30/30 which in a .308 is about as perfect a bullet as you can get for an animal that is quite a bit lighter than a Labrador Retriever!