The TL866 supports those devices and will do the job easily. You can edit data within the TL866 software. Just read the device, save a copy of the data, then edit as needed, and write back to the device. The editor is nothing special but it works. I like that it will highlight all changed bytes in red so that you know what you have changed.
The TL866 is very inexpensive and has a big supported device list. Yes, some 'professional' programmers will have broader device support, but will also cost at least 10x as much. If you buy a TL866, get one with many adapters for various PLCC and SOIC chips. Make sure you get at least an 8-pin SOIC socket, I find that I use that one quite often for EEPROMs.