We would still have to house and feed them during the screening process on top of all the other administrative costs to carry out these screenings and deportations. The camps and holding facilities where they keep illegal immigrants awaiting trial and deportation are rife with mistreatment and poor living conditions. I can't image the refugees would fair much better. Why should we be burdened with the expenses of that to allow cowards and ideological enemies into our lands?
The costs actually aren't that large if done correctly, take Canada for example which has already taken in 25,000 refugees and plans to take in another 25,000 before the end of the year. Canada has used some pretty extreme screening measures, many of which could probably be eliminated for cost sake, and has done it all outside of the country, meaning we haven't housed them or fed them ourselves. If the US were to take notes from Canada on this they could do it for relatively cheap, saving the lives of tens of thousands for a tiny tiny fraction of the cost of war. Screening can easily take care of any potential dangers due to ideology as I've said.
And these people are not cowards, most of them simply can't kill other people or think that the war will not help no matter who wins. I'm not even going to get into a debate about conscientious objectors but advocating for their execution is more than extreme.