How do you calculate the spellcasting ability modifier in 5e?
Your spell attack/ spellcasting ability modifier in 5e is equal to your proficiency bonus + Wisdom modifier. It equals +4 in this case. So let us interpret it simply. Suppose you cast a spell that needs a spell attack roll. You need to roll a d20 and add four to it. 5e Spiritual Weapon states damage is “1d8 + your spellcasting ability modifier”. Suppose the spell asks to add a spellcasting ability modifier. So do you prefer to add the modifier (wis or int) or the full spellcasting ability (wis + proficiency.) You only implement the modifier. Spiritual Weapon 5e states the damage as (per PHB p. 278):
1d8 + spellcasting ability modifier
The spellcasting ability modifier in 5e is the ability score modifier for that offense – Wisdom, at your aforesaid +3. You may increase the damage dice. It is through casting that at a higher spell slot. Suppose you use a Level 3 or higher spell slot to cast the spell. Then the damage will extend by 1d8 for each two slot levels over the 2nd. That is diverse than how you prepare a successful attack. The use of proficiency bonuses never applies to damage but only to attack rolls.
Read: Hand Crossbow 5e
Determining a successful attack involves rolling a d20 and summing both your ability modifier and proficiency bonus. Page 193 of the PHB has the written rules.
Spell attack bonus
The spell attack bonus hits the target, giving you a bonus of your spellcasting ability modifier (Wisdom in your case) plus proficiency. That’s the “melee spell attack” referred to in the spell’s description. If it hits 1d8 plus your ability modifier (Wisdom again), expect the damage. Therefore it is “only” 1d8+3 for you now. The Spiritual Weapon practices your bonus action to make a move/attack. It means that you can attack with it and your attack, or attack with it. You may cast a spell or manage an item. And all happens without the need for concentration to sustain the spell.