Just a note: I'm been going trough the wiki and discovered something.
According to the new movie, the Thing cannot replicate sythetic material (such as teeth fillings or even piercings).
This means that the tech-priests and anyone augmented cannot be simply taken over.
IIRC the thing in the movie was trying to build a craft to take itself out of the arctic.
Was it stated that it only wanted to get out of the artic or that it wanted to get off-planet?
Because if it had the capacity to build the spacecraft from spare parts, it is possible that it has a cell-level memory of some kind and that it has a mission beyond Earth. A mission it wanted to return to.
By being immune to chaos, by infecting a host, the thing makes that host also immune to chaos. Thus spreading the Thing greatly limits chaos.
1. The Thing is immune, and when it enters a host, is safe to assume that only the Thing-parts remain immune. Chaos infects people's soul after all and once the Thing takes over, it has effectively killed the host to replace it with Thing-self. It is said in the OP that Chaos cannot make the Thing it's servant, not that it projects an anti-chaos field like a Blank.
2. Except when infected by the Thing, you become the Thing. The host is killed and there is no known cure or prevantative.
3. This assumes that the Eldar have the technology to safely contain the Thing.
4. The Eldar have to know what the Thing is and have to see its chaos-immunity.
At best, you could try using it as a biological weapon (which I am guessing is what you intented?) but that then boils down to the target's resistance.
Against Chaos beings, it is unknown whether the Thing can be any effective. It is likely that Chaos will simply overwhelm the creature, no matter what form it takes.
Also, being immune is not the same as being lethal to. It is possible that the Thing cannot mimic Chaos beings and hardcore Chaos followers because it does not have the taint of Warp.
Even if it manages to fool Chaos beings not looking closely enough and it manages to infect and take over important Chaos worshipper, the lack of taint may be still a giveaway.
Therefore, attempts at its infiltration against Chaos will result in failure, making it a bad bioweapon.
Against anything else, as a preventative measure to deny more followers for Chaos, a more simple and reliable bio-weapon may be better. Or even, you know, a nuke.
The Thing is intelligent and has its own intentions, making it somewhat unpredictable. It may take revange on the Eldar or do things that the Eldar did not want it to do.
Because it knows at least as much as you do about the universe you just dropped it in and it's not a fucking moron.
If it wasn't the moron, it wouldn't have transformed itself into an acid-spitting tentacle monster when it was isolated with the dogs. Instead, it could have slowly infected every member of the team, one by one (it just would have had to bite the dog handler to mask an injection, wait til dog handler is Thing and then infect everyone else). Hell, if the other dogs were OK with it, why couldn't the Thing just slowly infect them too?
Either it has limitations on its virulency or it's stupid.