Something is sticking or maladjusted- it shouldn't take excessive flow to start an ignition cycle.
When you say it's working correctly, are you seeing an increase in flame height?
These beasties often regulate the output temp when the flame can't keep up by restricting the flow- it's conceivable that the internal mixing valve is sticking or restricted, but I'm thinking that's what the rebuilt kit should have taken care of.
If you have hard water and it's been in service for more than a year, it's possible that it's liming up on the inside of the heat exchanger. To de-lime it you have to give it a good rinse with white vinegar for an hour or so, which requires un-plumbing it from the water system. A small submersible sump pump in a 5 gallon bucket with 2-3 gallons of white vinegar running for an hour is usually sufficient. Turn off the gas, disconnect the potable plumbing, and plumb the output of the sump pump to the cold side (or hot) of the unit, and a hose hooked to the other side returning to the bucket, letting it run for an hour or so. (In hard-water country installing them with isolating ball valves and hose connections for frequent deliming is usually a good idea.) Don't put in anything stronger than vinegar or you'll likely damage the heat exchanger.