While I am an attorney, this is not legal advice. You should always consult a lawyer licensed in your jurisdiction and familiar with the relevant law before making legal decisions.
That being said, it's worth a shot, depending on the state. You should probably get at least one baseline legal opinion for your process, because some states have anti-harassment laws about sending unsolicited invoices.* Have an attorney draft you a form letter to go with your invoices that meshes with the invoice terms, etc. Then you can do it yourself or even have an assistant/secretary/whatever print and mail them, if you have one.
One thing that might be useful support for this tactic is to put language like this in your IPTC tags:
"The paid invoice for this photograph contains the grant of usage rights and duration of the license. All other rights including copyright remain the property of (Entity.) Usage without prior invoice constitutes agreement to pay (Entity's) then-current rates for all applicable usage types."
Where (Entity) is your corporate entity, or else just your name. Then you have a two-pronged attack. If the info's still in the tags, you've got them there, and if it's not, you claim deliberate removal of copyright information - which is subject to serious sanctions and damages.
Would it work? Hell if I know. You'll still have the off-shores, the fly-by-nights, and the lone pamphleteers, but I think it's about your best shot for an inexpensive enforcement attempt.
M
*Some of the really radical anti-government/no-tax types were using a tactic for a while where they'd "copyright" or "trademark" their names and then send huge bills to law enforcement and court personnel for infringement when they'd put the names on court documents or other legal papers. Creative, I'll give them that, but it did cause a few places to pass laws making it illegal to bill someone when you had no prior business relationship with them.