Do a google on Willem EPROM or look on e-bay.
There's always a lot of discussion about the Willem programmers (Willem (=William in Dutch) was a Dutch guy who once designed a cheap and simple EPROM burner, the design has been around and about and improved ever since).
They are dirt-cheap, you need to hook up a PSU yourself and some say they work great and others say not.
I have no experience with them.
A professional burner is a lot more expensive, you can go up to 1500 Euro's for the most elaborate one (but this one does them ALL).
You can also search KLOV forum about them for loads of info. Usually burning/Reading ROMs is something you will do occasionally. I've read (EP)ROMs a couple of times at work. They always ID-ed correctly so far. Haven't done any actual burning yet.
If you want to buy one, make sure it can handle the old EPROMS like 2716 etc. etc.
Again a question about the ROM testing screen. Does it simply show all the ROMs and stay red, does it go through the check of all the ROMS ? ( I do not know how this works, just want to know if the check stops at the first ROM or goes through each of them) ?
About the Battery: Hey, I'm only human I COULD screw up. But I doubt it in this case

By the way, here is the online version of the ROM identification tool (I love this because it's SO easy to use AND is available everywhere there's internet...):
http://www.coinop.org/RomId.aspx