How is the 1284 clocked?
Full swing?
The symptoms sort of fit the clock bug with the ATmega1284 processor.
If you have crossed paths with the bug the solution is to use full swing oscillator.
It really sounds like something with watchdog at first.
one chip works with or without the bootloader and the other chip only works if there is no bootloader.
I would focus to a difference. Try to read out both MCUs FLASH, EEPROM, fuses and compare it, also label could be interesting.
I remember something about ATmegas. Atmel removed full swing because of noise or something, but it was not a case for 1284P.
I don't think the crystal matters too much. (Though, that one does have nice specs.)
While the clock will sometimes oscillate without the capacitors it is important to include them. Does your circuit include the two load capacitors?
The fuse settings are what's most important. Those have to be set to full swing.
Ah. Now I get it. "Full swing" is a fuse setting not an attribute of the crystal. "Low power" is the alternative for an external clock source like a crystal. You do not want "low power". The original Arduino fuse settings are "low power". You need "full swing". Section 7.4 Full Swing Crystal Oscillator has details.
Not clocking. Got it.
Are you still dealing with the problem?
Fuses look ok. Only high:3 which is EESAVE. If you want to preserve EEPROM content you can use high fuse 0xDE.