If you have the low byte fuses set to CKSEL0 = 1 and SUT = 11 then you already have 16K clock cycles plus 65 ms delay between power up and reset. See table 5 on page 25 of the Atmel datasheet. I rather doubt your relay fix does anything, except by accident. But there may be some high current spikes drawn by either the LCD or the processor, or both, on power up, and maybe that is tripping the 7805 into a fault condition. I never liked 7805 regulators, some do strange things.
Its your ( or your bosses ) choice, but I would not release such a device for production until problems like that are fully understood and solved.
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