Agree with PPs. I've never been invited to a housewarming party, at least that's not what they called it. It's always "Hey, we're having a BBQ. Come see the new house!" The only "gift" I'd bring is alcohol or food. I'd feel pressured to buy an actual gift if they called it a housewarming party, and I'd have no idea what to bring.
But no, I wouldn't side eye you for inviting me over for a BBQ or something, and then a month later sending me a shower invite. Two separate things.