Can a Soldier be chaptered out for wanting to x-fer from a flight company to a maintenance Co if he has a down slip and can't fly?

In short no they can not chapter you for a down slip. Fought this battle both ways in my time as platoon sergeant of a flight company. Flying is a privilege not a requirement. If the soldier is holding a flight slot by MTOE and he is permanently disqualified from flight it is in the chain of command's best interest to fill it with a qualified soldier. When I was serving there were zero slots in the flight company for non-flight Crewchiefs. They were moved to the maintenance company and other candidates are swapped into those positions. Additional duties should not preclude this move. If the company is I that bad of a position that this soldier has 9 additional duties and they do not have a replacement for any of them that is the sign of a bad command team with no forethought or preparation for the future. Anytime a slot came open due to PCS, ETS, or other menial reason the alternate should move up and two new alternates should be found.

With that said if the down slip is temporary the flight company will probably hold onto a soldier. Especially if they are a good Crewchief and leader. This situation is always clear as mud as it is always a case by case situation and the commander has final authority.

Can they chapter him? Maybe, based on more info. A commander doesn't have to authorize a local transfer, he can simply refuse to sign a 4187 transferring the SM and he isn't going anywhere. Now, maybe the SM is really good and being punished for being good. However, there is always more to the story and it is important to see who said what and on what grounds. I wouldn't go to IG yet, as that irritates people sometime. Use the NCO support channel and get a little more feedback. Maybe they can start training others for some extra duties and eventually they have to prepare to lose any individual Soldier. Later based on feedback going to IG may be a really good idea.
I'm not aviation, but if you cannot fly you cannot do you job and there are legal ways to end Soldiers service when they cannot do their job or cannot deploy. The SM probably has a medical profile and based on the condition when appropriate a permanent profile may force a SM into the IDES. Or maybe you have a profile and haven't done an APFT in a long time, which is stalling your PME attendance. In that case a bar to reenlistment would be relatively easy. Those are just two examples on limited info.

It's been a while, but I'm pretty sure that things haven't changed all that much. Especially AR 600-106. It's fairly clear: 1-4d d. "Unit commanders will - (4)Ensure the number of personnel in crewmember flying positions does not exceed the number of authorized nonrated flight
positions documented in TAADS. Excess, overstrength, and dual-slotted personnel are not authorized crewmember flying status"
The idea of "authorized' is pretty clear as well: 2-6. "Medical Qualifications a. Nonrated Army personnel covered by this chapter must meet class III flight physical standards per AR 40–501."
If his flight status is terminated as per para 2-9 c. (4), then keeping him around in a flight company is really up to the Commander. If he wants to eat that slot with someone not qualified to fly, then it's his prerogative and he will have to answer to higher why he chooses to do so, as it begins to impact the overall readiness of his parent organization.
Of course, this is with knowing that which you have indicated. Not having an idea of the MTOE and a lot more than should be circulated on RP, that's the best I can offer.
*Can* they do that? Sure. The CO can pretty much do whatever he thinks his career can handle. Is it advisable? Probably not. It's really up to the unit Standardization Pilot/Instructor and the NCO support channel to make sense of the situation.