Is there a device/technique, whereby under eye bag fat could be melted away and the skin tightened?

There is no machine that can accurately melt or dissolve the lower eyelid fat.  Most of the machines that do fat  "melting" are pretty big - so can't get them focused on the lower lid. This area is also pretty unforgiving since the skin is so thin.  If the fat were to melt more in one area than another you would appear lumpy.  There are lasers that can help to tighten the skin of the lid, and that will sometimes help to pull the fat in.     

Bottom line is that none of them are sufficiently reliable to date.  You are basically inviting the surgery to experiment on your bags.  Generally surgery is a more reliable approach.  Interestingly, I have been very successful for certain types of bags by chemical peeling the lower eyelid.

Melting the fat deep to the lower eyelids can be performed with electro-cautery, Laser or radio-frequency devices.  The risk is damage to the infra-orbital nerve which gives sensation to the central cheek and upper lip.  Please consult in person with an experienced Board Certified Plastic Surgeon prior to making treatment decisions. Cheers.