But in conclusion, Heavy Armored Monk is feasible at lower levels, but gets kind of weak as you go up.
I think it stays pretty feasible, but you want some specific choices. It won't be high damage, but it is high mobility and with Fighter you can make it more survivable and "tough" than probably any other non-caster multiclass combination.
Far more survivable if you allow the 2014 subclasses (i.e. Long Death).
I mean think about this at 15th level with the right feats. fighter, Defense, Dwarf, Tough Feat S20, D14, C12, W16, 140hp, proficiency in all saves, Evasion, Heavy Armor Master, Shield Master
You have AC of 21 (not including magic), you can pretty much grapple and prone an enemy at will with 3 attempts to fail 2 saves, and hit another with a SAP weapon or alternatively take dodge as a bonus action at will giving everyone disadvantage against you
A typical attack action against 2 enemies goes like this:
Hit someone with your Longsword, then use your shield to knock him prone
Extra attack with someone else with your longsword (making his first attack at disadvantage), sheath sword
Flury of blows:
Grapple the prone guy
Attack the not prone guy, if you hit 5 force damage and try to stunning strike
One more Flurry of blows if you miss on the attack or the prone guy makes his save against either the shield bash or the grapple.
This is 2 ki used, which at this level means you can do this essentially at will.
Guy on the ground is now attacking with disadvantage because he can't get up. If he made three out of four saves he can get up but he is still attacking with disadvantage on his first attack due to SAP.
Guy not on the ground can't attack if he is stunned. If he made the save on stunning strike his first attack is at disadvatage.
The first attack that gets through (probably with disadvantage) has damage reduced by 22 .... and another 5 if it is BPS. Any other BPS attacks that get through lose 5 damage as well.
If you are fighting a huge bad guy you can't grapple you can still prone him with a shield bash, attack again with advantage using your attack action, potentially switching to a Monk weapon to try stunning strike, and then use dodge as a bonus action.
Playing a Long Death: any time a guy near you dies you gain 18 temp hps and in the extremely unlikely case that someone actually drains your 140 hps you use a ki and stay at 1 .... even if you are incapacitated.