Didn't the vet check his ears, eyes, mouth and limbs? I would think they'd be a good place to start. Eyes and ears in particular - eyes because inside the house, in dimmer light, he could be having vision issues - especially as he's not a spring chicken - and ears because its easy to miss ear infections and in some cases the first sign of an infection is a grumpy/snappy dog. His mouth should be inspected for broken teeth, lodged sticks, abcesses etc - especially with him not eating normally.
Other than that, by giving him attention and trying to distract him and work out what's wrong, could you be reinforcing the whining?
Any attention is worth having for many dogs, so if him whining gets your attention and provokes activity, it might just be rewarding in itself. Alternatively, it could be that all the activity when he whines makes him more anxious, hence more whining.
I know you said you've tried ignoring it, but you need to do it consistently for a while for him to get the message. If you think he needs to go out, get up and let him out, but calmly and quietly without any fuss or chatter, no "what's up Archie, do you need to go out ......?" etc.
One idea is to give him a non-attention signal eg hanging something on a door handle in his line of vision and then completely ignore him, starting with a few minutes and gradually building up the length of time. Eventualy he'll learn that when the signal is hanging up it means no attention and hopefully will start to relax and maybe just settle down in his basket at those times.
If he's not doing it when he's out and about and distracted and is happy to run around as usual, then that makes it less likely that arthritis pain is at the root of it, but doesn't rule it out completely.
Appetite changes are often an indicator of a problem, but again, could be anxiety or physical.
I think at this point, I would try the anti-inflammatories and if the vet hasn't done the other checks, I'd go back and ask for him to do them. I would also make enquiries for a behaviourist, as they will be very experienced with dogs and may be able to spot something both you and the vet have missed, be it physical or behavioural.