Do you have any spares lying about that you could troubleshoot with?
I generally have an old bulb, past it’s life but still working, that I can test a ballast with. I also always have a spare timer, just in case.
I personally would not run a 600W bulb directly from a timer, timers are notorious for having poor contacts and a HPS light draws power in a different way from most devices which is harder on the contacts of your timer.
If you get a 4 way extension lead and carefully strip some insulation from a couple of inches of the cable, leaving the three wires inside with insulation, then break the live wire and put a relay in there and connect it to the ‘normally open’ contacts then have your timer supply power to the coil when it turns on then you now have a ‘contactor box’ that cost you about £20 and, depending on the relay you use, can switch up to 2.4KW of power.
You can get physically larger relays with a clear case so you can visually inspect the contacts as well.
It may be that the bounce on your timer has affected your ballast or bulb. When a relay switches, the contacts bounce for a bit, turning the power on and off rapidly for a short time. This can sometimes cause issues especially in cheap timers.
But to troubleshoot your problem, I would try the ballast and bulb on their own. If they work fine it suggests the timer is not working. If they do not it does not mean the timer is good though, sometimes a failure in one device can cascade and lead to other failures. Test the timer with something that draws a similar amount of power as well.
If possible try each individual component in isolation from the others, if they all work see what the combination is that makes it fail.
I have had cases where a timer went and took other items with it leading to multiple failures at the same time. If your contacts are on the way out, it is possible that your timer is sending voltage spikes down the line although this is rare.
Just noticed you do not say if your ballast is digital or not. If it is not then it will draw a lot of power to start and even when on, as it is an inductive load, it should be treated as double the rated amperage for timer/relay purposes.
How new is all this kit? Is it under warranty?
If it is, I would take it back to the shop and tell them you need a replacement the same day for obvious reasons. A decent shop will realise how serious your concern is and, if nothing else, supply you with a replacement until it is returned or replaced.
My tests would be, try a different bulb straight to the wall socket, try the timer with an equally demanding device, try a different ballast. You say your light sometimes does not turn on even at the wall socket, is this with both bulbs? If only one it might be your ballast is failing, or just one bulb.
But if in warranty, I would take the ballast, and bulbs to the shop you got them from and get them to test it all. They may well have another ballast to test your bulbs, and other bulbs to test your ballast. Take the timer too and have them test that. That would be the best and quickest solution.
If everything works fine away from your home, test your mains for voltage spikes…