Your houserule is not overpowered
It is very much needed just to keep the Monk/Wizard competititve.
Compare the damage he does to a 5th level Monk
Both versions are using a Quarterstaff, they have no reason not to.
To show that the player is not only behind because he did not have an ASI yet (to boost Dexterity for example) the example L5 Monk does not have it either.
With Extra Attack, the damage would be 2x(1d8+3)+1d6+3 = 21.5
Monk 3/Wizard 2
With Green Flame Blade, the first target takes 1d8+3+1d8+1d4+3 = 17.5
As it was the accepted rule of thumb on the now defunct WotC forum, we calculate with half the damage the secondary victim takes (it might not even exist, and killing the primary target is more important than spreading damage). So the final damage is 17.5+(1d8+3)/2 = 21.25
Even if he gets to Extra Attack, it can not be used with Green Flame Blade, so this is just a temporary patch anyway.
Admittedly, the wizard levels give it more versatility, but actually make it weaker. And do not forget, that he is as MAD as can be because the monk needs Dexterity for attack and defense, Wisdom for defense, Intelligence for casting, and Constitution because everyone needs a decent value in that ability score.
This is an exception
This only works because there is hardly any synergy between Monks and Wizards. A Cleric of Arcana (SCAG) can use the cantrips, but based on Wisdom. For them this house rule would be overpowered.
Monk/Wizards are on the weak side of multiclassing, so giving them this minor advantage just helps them not to be too far behind.