The cash you need for the day can be divided — some in your pockets and some in a wallet (see How to Prevent Serious Cash Losses with a Fake Travel Wallet).
If your fakewallet is also in that basic purse, but your primary travel cash, documents, and passport are in your money belt worn underneath your clothing, then you can hand the purse and fakewallet over without too much remorse.
Some travel experts recommend putting in some old cards — outdated library cards and used gift cards work well here — and even some fake money into a cheap wallet.