iPhone 4 Battery Problem
That's not a battery problem unless it powers off while you are trying to connect it to a computer / powered USB hub to try and pack up the photos.

Taking it to Apple for service won't get the data back as they erase the device if it gets swapped. If the battery indicator is not showing, it could be a bad battery (even some new parts are DOA), the connectors failing or the control circuitry on the main board is faulty.

Worst case, get a quote from a data recovery service and understand if they charge if they cannot get back what you want. Be careful - some say if any data is recovered - be sure to specify you only want photos if that's what you're considering paying for.

If you have a Mac - check out PhoneView to restore the photos. There are lots of other apps, you can search them here for specifics but you might have passed the window where you can get the photos back.