Hey Christy! 
Well, as you can imagine, this is a security risk for a Member Group to be specified via a form field, so this is currently not available. 
Although, after some consideration we’ve queued up this option, but will include a special parameter along with it that will specify which Member Groups are allowed to be selected through a user-selectable form field, or in your case, a hidden field.
Will probably look something like this:
{exp:user:register allowed_member_groups="members|pies|pants"}
content
<select name="member_group">
{user_member_groups}
<option value="{value}">{value}</option>
{/user_member_groups}
</select>
{/exp:user:register}
We just need to crack the whip on Mitchell for this, haha
Otherwise, yes, only the Invite Key will do this 
