A sticky dilemma.
I'm part of a team based in USA that produces a virtual world software for remote business purposes. The businesses that use us are our Clients with users from all over the world (and expanding), but primarily in the USA. Our software makes use of customizable human avatars to use in world for each user. We have gotten requests from one of our biggest paying Clients and approval from boss to include religion based avatar clothing options (yamulkes, headscarves, skullcaps and turban head coverings currently, potentially garments too).
As our software is used for business, most people want to keep their real world likeness, which may include some of this clothing because it is a part of their identity. Since this is such a sensitive topic on all sides involved and we are in a politically charged climate in the USA, clearly we don't want to offend anyone because they all pay us. In my opinion, even if this request was deemed as a reason for loss on Client's part, it will still be our company providing the service that will be affected primarily. As an emerging company we can't afford to lose users or current/potential Clients over something unrelated to the core mechanics or hardware requirements of the game.

How do we put it in the avatar creation menu? Keep it with the other head coverings (so not to upset/offend the religious wear users via segregation) or separate it (to protect from accidental abuse of said garments from ignorant users and offend everybody)? As difficult as it would be for us to do (right now), do we only allow access to certain users? Would that be going too far to request information such as this from users, or for them to have to volunteer it for access?

How do we talk about it with the client? When the concern was brought up, they warned us to be careful about using the term "religious wear", so we switched to the more broad "cultural wear", in which they again implied even that term might offend in discussion (because Texas users (very many) would get mad about their cowboy hats not being treated as culturally significant...) and client tactfully avoided telling us what they want us to call it themselves. How do we have a productive conversation though they put out a controversial request and are not willing to speak confidently on it's behalf?

It seems your questions themselves state the solutions. 

2 hours ago, x3ph3r said:

How do we put it in the avatar creation menu?

Starting the label with Religious should be a cue to most people. Maybe adding  "Meaningful" or "Sensitive" would add a further note to the user to the user.  If I saw the branch in the tree was labeled "Religious & Culturally Meaningful" I'd pause a moment before doing too much with it.  The label immediately communicates to me that it contains stuff people have fought wars over and I better not throw it around casually.

Then I'd cluster everything inside that.  Perhaps on organizational tree by religion, another by geography, and allow for significant overlap between them. Then under that your regular organizational structure by body area or whatever.

There are many items of clothing that one group considers sensitive where another does not.  Scarves and shawls and caps and hats are casual in one culture, but may appear similar to one that another group considers part of their vestments. That could cause some confusion, and could allow for items to appear in many places, both in the religious and the non-religious clothing groups.

2 hours ago, x3ph3r said:

How do we talk about it with the client?

You described it here well enough.  A label of "Religious & Culturally Sensitive" is clear to me that you aren't talking about cowboy hats, wooden shoes, or local clothing.

If someone asks, you can describe it as you did in this post. It means clothing that is significant to various groups and should be used with respect and forethought.

But as mentioned above, that doesn't mean anything outside that cluster can be used freely.  A lacy scarf is meaningless to one group, or part of religious garb for another group.  The person using the clothing needs to be aware of the context.

You can't be sued for including religious symbols in a "video game" (I know you are more of a business interaction type software but I'm going to use games for my example).  Ok TECHNICALLY they could, but they would lose. They can be offended though.

I would say the best way to avoid offending people is not to not separate it into a special category.  For example just put it under head wear. I find that most people get offended not by being included, but by being made to feel different.  This is just my opinion though. 

I would recommend that before you include anything you ask someone from that religion if the religious garb is considered a publicly religious item or a private religious item.