Okay, I've figured this out, in case someone ever wants the information. Push the latch tab all the way back and rotate the key to the right. The button will then come out. Installation is the reverse - push the tab back, rotate the key to the left.
Someone has been selling a small mountain of NOS Briggs & Stratton housings (without button) on eBay. If your latch is broken, but you want to keep it keyed to match your ignition, this might be the way to go.
