Pretty sure there are modded BIOSes with the whitelist removed, they exist for laptops.
These wifi adapters should work https://github.com/morrownr/USB-WiFi
in order to get something compact, personnaly i wouldn't go with a usb dongle, nor use the internal mpcie slot. i'd rather use somehting like the netgear wax214 placed on the hood of the m720q.
Seems very expensive...
The official I350-T4 from Lenovo FRU 03T8760 / part number 4XC0R41416
can be found for this price is you search on google
https://www.lenovo.com/us/en/p/accessories-and-software/thinkcentre-and-thinkstation/thinkcentre-and-thinkstation-cables-and-adapters/4xc0r41416?orgRef=https%253A%252F%252Fwww.google.com%252F
and they give the baffle + Riser.
For this price i would even go for a multigigabit X550-T2 10/5/2.5/1G 
original from Lenovo :
it's too long though, 21+ cm.
Why not use any dual radio router, set up one radio as a wireless client, 2nd to retransmit the signal.
Make sure you route the traffic, and not only retransmit.
Any openwrt device with two radios should be capable of doing this, if you want something really small, look at gl.inet.
Obviously you only need one radio, if you're not retransmitting.
Or, add an USB wifi stick to the M720.
i don't get it...why don't you just connect your laptop to the Free Wifi ?? 
For these exceptional situations why not take a gl.inet as indicated by Frollic.
it will be less efficient but you don't travel every day? do you?
for those interested, some I350-T4 available for 50€ :
i got mine from this seller, mine was branded CISCO, but this one looks exactly the same.
They are sold as "used" cards, but i received a brand new one.
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