This sounds like a pretty dreadful experience and I cannot understand why Qantas wouldn't respond to your complaint. The emails to customer service are often erratically answered and I agree that it is just not good enough. I haven't dealt much with the business check in at LHR recently but have always found the F check in staff to be OK. Qantas has a mailing address for head office :
Customer Care
Qantas Airways Limited
Level 5 -Building A
203 Coward St
Mascot NSW 2020
There is also a fax number :
+61 2 8222 4700
I would hope that they would respond to one of these channels of communication.
Good luck