Adding profiency to your damage roles is not even remotely balanced***. Your to-hit chance between unarmed vs weapon attacks is the same, so to compare balance for #2 between regular monk and your homebrew all we need to compare is average damage. We can assume hit rate is 100% for the sake of comparison because of this, but of course the damage values I will post here are not real world ones, and are only accurate relative to each other. So for example, at level 1 with a quarterstaff, we hit with the quarterstaff once and hit with our unarmed once, dealing 1d8 + (whichever ability mod) for the first and 1d4 + (whichever ability mod) for the second. With your homebrew, we would instead hit for 1d4 + 2 + (whichever ability mod) for the first attack and 1d4 + 2 + (whichever ability mod) again for the second attack. As you can see, for the sake of comparison we can remove the ability mods because they remain constant, which I will do so in the following formulas.
level 1 with quarterstaff:
1d8 + 1d4 = 7 average damage
level 1 with proficiency added to dmg rolls:
1d4 + 2 + 1d4 + 2 = 9 average damage
This homebrew increases our average damage by 3 at level 1. If our ability mod was +3, we'd have 13 average damage vs 15 average damage in these two cases.
level 1 with quarterstaff (using flurry of blows):
1d8 + 1d4 + 1d4 = 9.5 average damage
level 1 with proficiency added to dmg rolls (using flurry of blows):
1d4 + 2 + 1d4 + 2 + 1d4 + 2 = 13.5 average damage
An even bigger increase. If our ability mod was +3, we'd be at 18.5 damage vs 22.5 damage, which is almost a 25% increase. What happens at higher levels?
level 5 with quarterstaff:
1d8 + 1d6 + 1d6 = 11.5 average damage
level 5 with proficiency added to dmg rolls:
1d6 + 3 + 1d6 + 3 + 1d6 + 3 = 19.5 average damage
An even bigger increase. Originally I was musing that #2 has no disadvantage anymore at level 5, but it turns out that it's not so. Much later, at level 11, our martial arts attacks are as big or bigger than any monk weapon's base attacks, so the weapon stats no longer matter and #2 becomes strict upside. But hey, TIL that monks have access to 1d8 weapons! This DPS discrepancy will grow as the proficiency bonus increases.
*** edit: I did not realize this was replacing a subclass, lol. I guess it might be balanced, then, at level 3 forwards when other subclasses get their effects. I'll leave all this math here anyway, but it's worth noting in addition to not getting anything until level 3, monk subclasses like Way of the Open Hand usually grant abilities or conditional bonuses. Way of the Open Hand lets you do a thing when you use flurry of blows. Your subclass gives the player a pile of properties (some drawback, but most advantageous) that are always passively on. This is interesting as a break from tradition, but it makes it hard to balance. How much passive always-on damage is worth being able to have Way of the Open Hand's effect on flurry of blows? Who knows? I don't know.