It turns out the Hillary had already done what I'd asked her to do...last month. According to TPM, she submitted a bill which would ban expenditures on any SoFA with Iraq which wasn't a formal treaty.
Great. Only one problem: her bill has only one co-sponsor, Sheldon Whitehouse of Rhode Island. Hey, Senator Biden? Senator Dodd? Senator Obama? (not to mention Senator Reid) -- why haven't you cosponsored this?
Update (From the comments) SoFA is shorthand for "Status of Forces Agreement". Sorry for the obscure acronym.