Jim is a respected member here. He is trying to explain to you why your idea won't work. You clearly don't want to listen, just want someone to tell you to do what you already intend to do. That's not the point of this forum. We're here to tell you how to do it right, and tell you when your idea is wrong.
You're backcheck/extra pitch/etc ideas will make your problems 10x worse. You're going against not just code, but basic fluid dynamics.
You need to get a vent into this setup. Use an Air Admittance Valve if you absolutely have to, but don't just skip the vent. We're not talking about "code scmode" here, we're talking about serious health risks. Just b/c something you do seems to work (which just tying into that line very well may appear to be "working fine," doesn't mean that its working safely.
Ultimately, the advice I would give, and this is also coming from someone who doesn't hold the code in particularly high regard but cares about doing things at least as close to "right" as possible, is that you either need to learn a LOT more about plumbing before taking on this project, or call a plumber.