Can immigrants be used to increase GDP without hurting GDP per capita?

I’ve done some research on immigrants, legal and illegal, and have noticed that most of the poor immigrants eventually converge in productivity and income of natives over time. Some studies I’ve seen actually linked immigration with higher levels of productivity because they introduce new ideas or systems, and build on the concept of economies of scale. The US has a long long history of immigration and integration, and yet we spend MUCH more money on policing them than integrating them, especially illegal immigrants.(side note, we spend more money on ICE than we spend on social services for illegal immigrants, don’t let people tell you they drain resources) Can we choose to spend money to better integrate immigrants at a massive scale, like a couple million a year, without significantly hurting short or long term productivity or gdp per capita?