Before doing anything else, you might consider performing a partial disassembly and cleaning of your burner, as referenced by DigaDo in this thread
. Panasonic recorder burners are notorious for prematurely "playing dead" when they get a little dusty or grimy, but after cleaning they often go on working fine for another few years. Of course, if yours is truly dead, cleaning won't help.
The Panasonic DMR-XW450 seems to be a 2009 model, while the DMR-EX77 seems to date from 2007 or 2008 (reviews on various websites conflict). Both are Australian models never sold in North America, in fact all Panasonic DVD/HDD recorders were pulled from N.A. after the 2006 EH55 series. So unfortunately there are very few AVS members with first-hand experience repairing these models: most of us can only offer a guess. Odds are the burners are not compatible, as the XW450 has ability to transcode HDTV and the EX77 does not.
The $300 repair fee quoted by Panasonic is fairly typical of what mfrs charge to replace the burner in DVD/HDD recorders, one of several reasons they fell out of favor. If you discover there is no feasible DIY repair, having Panasonic fix it may be your only option unless you can afford a complete new recorder (and Panasonic is still selling anything comparable in your country).