This is a common, real-world scenario, and we see it all the time. Agencies, in fact, are used to it: it's called payrolling.
If you find the work yourself - i.e. someone wants to hire _you_ but needs you to go through an agency - then ask for the names of three agencies they work with, and if any of them payroll. (They may not know the answer to that.)
Then call all three, tell them the hiring manager and anything else you know about the financials (you might know bill rate, or what you want to make, or both), and tell them you're calling all three agencies, and you'll go with the one that can make the best deal. That's a win for the agency (a little money and another person on the books with that client) and a win for you (in comparison to the $65/$130 deal). Factoring in things like employment taxes and such, you can assume that you'll keep ~90% of the bill rate. (So $65 would become $117 total comp: if they're required to employ you and pay taxes, offer insurance, etc., more like $95 to you. Look, almost a 50% raise.)
We're a small agency (~50 engineers), and we usually have 3-5 people we're payrolling because either the client called us and asked us to payroll, or we got a call like the one above and decided it was worth it.