I haven't ever played DnD as a monk myself, is it really that important for them to be tanky? CON-wise? Is it possible to dump CON and let's say leave it at 12 or even 10? Can we make it work?
Absolutely not.
You don't wear armor and you don't use shields.
Despite all of this you are going to be in the thick of combat.
WIS and CON keep you alive by providing you HP and AC.
DEX keeps you alive be providing you AC and damage.
I would not make a meelee character below 14 CON, but 14 CON is really not good, 16 is where it starts getting good.
10 or 12 CON is only acceptable for characters who very rarely enter meelee.
Monks also don't get many HP point on lvl up, so keep that in mind too. You need CON.