That's why I have three "hot" bank accounts.
One for ATM transactions/meatspace debit card purchases, one for bill payment, and one for cyberspace debit card purchases.
If your bank doesn't make this easy to manage, switch banks.
Your plan is flawed. Sure, you have three accounts but your comment "If your bank doesn't make this easy to manage, switch banks" implies they are all at the same bank. Which subjects you to many other risks you're likely ignoring:
1. Person with stolen card may be able to social-engineer access to other accounts or online credentials and thus access the other accounts.
2. Your bank may choose to do a "courtesy overdraft transfer" from you other account, to cover thief's new laptop and vacation.
3. An "unusual transaction" on the one account, if unusual enough, may trigger the bank's fraud-bots to put a freeze on all your accounts, at least temporarily. Some stupid institutions do "freeze everything, no messages" as an attention-getting attempt at reaching you, and no, they don't disclose up front that they do that, so you can't "switch banks" based on looking out for that stupidity.
4. Some dispute with a big-enough jerk person, company, or organization may lead to a lawsuit or garnishment against you, and nowadays many banks have an immediate "fire the customer" response to that action. Again, not something they disclose up front. Condo Board (HOA) from Hell got me fired as a customer from a "good local bank". Luckily it wasn't my only bank/bank-alternative.
A much better idea, if you want segregation of accounts between physical world use, online use, and billpay use, is to use three different institutions entirely, picking carefully both for minimal Banksterism and for free external transfer services.
For example, I have (US-centric because that's the topic):
1. A Credit Union membership, in an institution that pays 4% interest (yes, four percent I didn't drop zeros or decimals) on the first $500 in checking and separately on the first $500 in savings. Has totally free 2-3 day ACH "push" to transfer money to any other bank or bank-like-thing (such as a prepaid debit card with a "bank account number" and "routing number" or to "pull" from any other bank-like account. Only if I initiate it. Overnight for a $2 fee. Both checking and savings there to maximize interest, have their Debit MasterCard, have their Bill-Pay but have no current payees set up, deliberately do not have any actual paper checks and never have on this account.
2. A "checking alternative" account with no minimum deposit, no minimum balance requirement, from an online discount brokerage firm (I don't have an investment account with them, just this cash management account.) Has a Visa debit card no added fees for foreign transactions over the Visa conversion fee, full rebate of any ATM-owner surcharge anywhere in the world, deposited back next banking day, no ATM-use fees of their own. Has free printed checks and free check refills. Has free BillPay, free external transfers by ACH. Pushes to my other bank-like institutions typically arrive next banking day despite their saying it is 2 days. Pulls from other accounts usually 2 days.
3. A high-interest (as US interests rates go) online-only savings account with no checking, no bill-pay, no nothing but can be the ACH target for direct deposit from Elance, PayPal, etc. for freelance work, is a transfer source and target for accounts 1) and 2) at those other institutions, has its own ACH external transfer capabilities (typically 2-3 days on pushes, out, a couple more days for funds availability on pulls into it - so I usually push from the other accounts which makes it instantly available when it gets there). Also tied to an online purchases rebates cashback program (Upromise.com - oriented towards savings for students but anybody can use it and get the cashback rebates, no matter what form of payment used, into their Upromise account and then transferred into this bank account.)
An action by an adversary (legal or illegal) against account 1) cannot cut off my access to accounts 2) or 3) directly. Access to the Debit MasterCard / Visa Debit of accounts 1) or 2) cannot access account 3 at all. Account 3) can push to a prepaid Visa from an entirely separate institution if I want to do spending in the real or online world that needs a card, with good funds on that card in 2 days.
Much safer. I'm obscured here, not anonymously nym'ed here anymore, and I've mentioned these in fully-disclosed locations like G+ previously, so I'll say what I have:
Account 1) BECU credit union in Washington State ("Boeing Employees Credit Union"). Anybody who lives in WA, works in WA, "worships in" WA or is a near relative of anybody who does or did, can join, and then keep the membership relationship after leaving. I think if you've just changed planes in Seattle-Tacoma International and stayed long enough to get Ivar's Clams, you're probably eligible. becu.org. Can mail or scan paper checks.
Account 2) Fidelity Investments Cash Management Account (used to be called "My smartCash" and is nothing like the "Cash Management Account" concept of a unified investment+cash single-account that the big wirehouses like Merrill use that name for.) fidelity.com, look for the Cash Management Account thing. Some of the other online brokerages also have similar accounts with comparable benefits, so check Schwab, etrade, etc. Obviously they are hoping to create a wider account relationship but other than the home page promoting other products, Fidelity does nothing in-your-face to get you to open other accounts beyond this way-better-than-bank-checking account. Only thing you can't do is deal in cash. If there's a Fidelity branch nearby you can deposit checks in person. Otherwise can mail or scan. There may be balance minimums to be able to scan checks.
Account 3) Sallie Mae Bank Upromise-linked High Yield Savings. banking.salliemae.com and upromise.com. You don't need to have a student, you don't need to set up a 529 plan.
If you don't want anything to do with any part of Wall Street (even its distant Boston relation which doesn't do its own underwriting or create its own deriviatives for the most part), then you won't want any type of account like Account 2. But I consider that self-defeating nose-cutting face-spiting. You can opt-out of the FDIC banks to which Fidelity sweeps your balance, getting rid of the worst of the banksters, using the less-evil smaller and regional banks on the "Program Bank List".
If you don't want anything to do with a bailed-out government-sponsored-enterprise (GSE) like Sallie Mae (or like Ally Bank as another example) then you won't want anything to do with Account 3. But again, why not take advantage. And however rapacious Sallie Mae may be, to at least some extent your deposit is helping provide student loan money. Meanwhile you're getting rebates from Lenovo, Dell, Newegg, Expedia, cheap hotel chains, and lots more, and every time $10 accrues, it sweeps into your Sallie Mae bank account.
I don't do the exact same type of account segregation as the AC to whom I'm replying. But I do some logical segregation and optimization. Funds sitting for a while, into Account 3) where they get almost a percent of interest in an account that there is no way out of that isn't directed, online, by me. No checks, no card, not even a checking account number. If I'm feeling at all flush in these transitional years between end-of-corporate-career (not by choice, by Banksterism) and early-social-security, and have an >4-figure balance, I pack Account 1) BECU with a full $1000 (considering the two accounts, savings and checking with 4% interest on $500 in each, as sub-accounts of one mental-aggregate-account). I move money in and out of Account 1) Fidelity with planning based on what I need to withdraw for free via their best-to-use ATM/Visa Debit. PayPal/Elance payments go into either 2 or 3. No frak-up in or about any one of those institutions can screw me over in the other two. The rare US billpays I need to do, I usually would do from one of the two bill-pay enabled accounts, and I wouldn't put all payees in one account, again for distribution of risk.
Three accounts, at the same place, give none of these benefits.