Shiawase Kanmushi. It's a microdrone that looks like a small bug. It's perfect for spying because of it's size and that it actually is mistakable for an insect.
Cutting through walls can be very silent actually. They also have a spray foam thing that melts glass silently. Seriously, if "through a door" is how you think a stealth approach goes... You should really study up.
Why wouldn't an illusion work? Just because it may have an aura doesn't mean it's automatically spotted as an illusion, especially if it's an illusion of a living thing.
Alternatively if they know there's a mage, they can try and eliminate them before entering. This will get rid of the spirits. How to do that? I'm not going to spell it out for you... But there's ways. Drones with drugs. Silenced sniper bullets through walls. Your own spirits... Be creative. Take them out in a way that doesn't immediately alert everyone else, and if that is impossible... I don't know what to tell you. You can't take out a single person without being found out, so stop trying stealth and just work on a better mission style.
Finally, it's inaccurate to say a silent approach is worthless if you can't pull of the entire run without being spotted. Getting in before being spotted is a lot different than going through the front door-- Err, sorry. It should be anyways. HTR team times can make a big difference.
Instead of trying to avoid setting off alarms, set off false alarms. Use distractions. The bigger the facility, the more vulnerable it is to such tactics. Drawing some or all of the security to a point elsewhere lets the stealthy parts of the team go in avoiding what's left while not needing to worry about alarms being sounded. If they are spotted by someone, it should be by the time they're already in the heart of the complex near their target. This can be accomplished with bombs, spirits, drones, illusions, or a roided-up street samurai for the distraction.
What I'm trying to say is, be creative.
Try using disguises! Unless the spirit is somehow in a place literally no-one is supposed to be, in which case it's so far into the facility that it spotting the intruders is too little too late and a waste of money.