Sen. Mike Lee objected on behalf of Rubio, who was in South Carolina.
The legislation was stalled over a fight on an Internet tax ban.
Defense hawks are calling for extra funding.
The Senate is hoping to pass the customs bill Thursday.
Sanders had expressed concern about North Korea during last week's debate.
Sen. Chris Coons (D-Del.) said he will try to bolster funding in the appropriations process.
The bill sets standards for research being "worthy" of federal grants.
The legislation would require the Obama administration to crackdown on the isolated country.
The letter come as the Senate is expected to back new sanctions.
Reid calls the decision "short-sighted" and a "stunning move."