I can't think of any safety reasons that you wouldn't be able to use a rechargeable over a regular alkaline/whatever cell.

Most rechargeable batteries do have a lower internal resistance but if your product is shorting out anyway then you've got other issues.

The lower cell voltage shouldn't affect much either since the voltage of an alkaline battery will slow drop as well.

You get warnings about putting non-rechargable batteries into devices that are expecting rechargeable batteries but this isn't the case here.