I would hazard a guess that most here - myself included - tend to simply recommend the rounds we use, for no better reason than we use them. To that extent I wonder how much objective analysis has really gone into that selection? As a result we seem to have ended up with as many options here on bullet weight and make as there have been replies, with little if any real consensus.
The OP requested...
Anyone know a good 308 round to use on roe deer with as little damage as possible
I am not sure about objective analysis, but I can claim empirical observation and Google research in my selection.
I equated the OP's "little damage as possible" to little wasted or discarded meat...so the non-frangible copper / monolithic win hands down...it is the main reason I use them.
I did presume Max ranges of 200metres as none was specified.
The frangible lead core bullets (SSTs, Nosler BTs, Powershok, Super Hammerhead and etc) I have used have all been much of a muchness with the monolithic as far as effectively killing Fallow, Roe and Muntjac at the ranges I shoot, which are similar to yours...mainly sub 100metres.
The bruising (jelly) between the muscle groups from hydraulic shock has also appeared to be more to do with POI than bullet type, though the monolithic do certainly produce their fair share it...but at least with the monolithic you can just scrape the jelly away and use the meat.
If you follow the recommendation of the removal and discarding of all meat within 6" / 150mm of a frangible lead bullet wound tract, you have to waste so much more...that is almost the whole far shoulder of Roe if H&L / Hilar Shot.
You missed out my 130gr Fox from your list, and you could add the 115gr Lehigh Defense Controlled Chaos monolithic as they have an interesting variation somewhere between the 99% retention monolithic and the frangible...close to they shed three or four petals and at greater range / lower velocity they rely on expansion alone.
Alan