You need to go contract Direct if you want to be treated as an “independant” consultant. The Agency usually makes you sign something that makes it near impossible to contract Direct for a company they place you at. You need to use your network to find your own contracts. This is easier said than done.
While an SS-8 is pending, you can file your taxes as if you were a W-2 employee by filing Form 8919: https://www.irs.gov/forms-pubs/form-8919-uncollected-social-...
I did this in Canada (long ago) with a former employer who contracted me but treated me as an employee, and the Canada Revenue Agency sided with me, regardless of the contract text. All I've read suggests that the IRS is similar in this regard.
Keep in mind that if you do this while you're still employed, they are likely to fire you. The ideal thing to do here, if you can negotiate with the employer in good faith, is to have them void the previous 1099s and file back W-2s for you, so you can then file amended 1040s and get back your previously paid tax. Unfortunately, an employer who would do this in the first place is likely just trying to avoid payroll tax liability, so I doubt you'll see much success going this route.
At the end of the day, you usually can’t make a bad client “behave”. You either pick good clients or leave.
If you don't mind losing the contract, I'd discuss their options. Something along these lines:
"I don't think our expectations are lined up. I expected this to be a 1099 consultant position but you are treating it like it's a W2 position. Before the IRS gets all up in our business, how should be we fix this? Do you want to treat me as true independent consultant per IRS rules, have the subcontractor hire me on as a W2 employee and bill me out like we're doing now, or shake hands and go our separate ways?"
In reality, this is why companies go through contracting agencies. The agencies don't (really) offer true employee benefits but the client companies still get the advantages of a 1099 employee (easy to "let go" at the end of the contract).
”The IRS developed the 20-Factor Test shown below to help employers evaluate whether a worker is an employee or an independent contractor. No one factor on the test is more indicative of employee status than another. If there is a high number of “yes” marks, then it will most likely indicate the presence of an employment relationship, a high number of “no” marks may or may not indicate a contractor relationship.”
https://www.regent.edu/admin/busoff/pdf/20-questions1099test...