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I agree with your assessment that it's at the burner but as it's clearly causing me a problem I'm open to all suggestions as food for thought while I'm lying on my sunbed by the pool
I did check that there was a good flame at the burner so it might be the transformer, or the leads, but surely that would cause an intermittent fault with the initial firing as well. It fires every time, after the reset but still goes to lockout

Once the flame is ignited (even just to make my point, if it were to be ignited by a match!) the burner should continue to operate.
Shutting off a few seconds after flame is established suggests control box, but can still be photocell faulty.
Transformer wouldn't have anything to do with lockout once it starts the flame.
Got to check the oil pump to motor drive shaft and drives are not worn and slipping.
Only you said the oil pump is renewed, I would also say check oil pump hasn't a broken drive pin
 
You could run it on an alternative supply of fuel to eliminate any problems with the tank, fuel line, check valve etc, but like best says, not going to divulge exactly how I would do that on an open forum.
 

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