So here's the thing. The ID requirement is to make money laundering and in-person banking more difficult, BUT it will also make it impossible for you to declare that you didn't deposit the money. The IRS doesn't care how you make your money, only that you report it and pay taxes on it. You could literally put HITMAN as your profession and they don't care, but the prosecutor following the money would care, it's just a question of who is coming after you. As for getting loans, etc., they may ask for years of tax returns, so may be best to some up with a better title, like domestic relations consultant, or just plain consulting.
ATM deposits will absolutely prevent the ID issue, but to the bank's concern, would you give your PIN to someone else to deposit your money, and also give them the option of withdrawing your money . . . money that you might have trouble explaining if it was stolen?
If it's just you, and you just want to not show your ID, then ATM should be fine. As for will you be flagged, that is an algorithm question. Better to find out what the minimum balance for no fee banking is, and then spread your money across multiple banks (and keep a safe with a few months of living money, so you can afford to get your money back via the legal process if necessary).
As a side note, all this shit makes me sick. Politicians playing these games DO NOT get my support, this is a non-issue. I care more about gun violence than someone selling pussy.