My experience is that HD Writer won't work unless you use it to transfer your video to your computer. Perhaps something changes in those three little extra files during the process. It seems if video files get to your computer any other way, HD Writer will treat them as foreign aliens. It simply won't work with them.
HD Writer should work if you still have your mother's SD card. Put it in the slot on your computer. If you don't have a slot, get a USB card reader (about $5). HD Writer should see it as a potential source under the "Copy to PC" process.
If you don't have her card or it doesn't work, you need other software.
The latest version of Adobe Premier Elements is 10. It sells for about $75 and does many fun, amazing and useful things - including the new feature of making AVCHD discs for use in Blu-Ray players. $25 and a few evenings at Lynda.com Training
and you will have a good working knowledge of all it can do.
Reading this forum over the last year, the think the most frequently mentioned "freeware" program is "MultiAVCD". I've not tried it and don't want to.
On edit: I re-read your question. You said this is a one time project. I think the PS3 will play .mts files from a plain data disc. Try using Windows Explorer and copy them to a disk like you would if you were backing up word processing files. I don't own one, but I've read where the PS3 will play .mts and .m2ts files like many Blu-Ray players will.
Good luck and have fun!