I2C is by nature a Multi-Master Bus.
Any device on the bus can be the "Master".
And the same device can also be a "Slave", it just needs to switch back and forth between the modes.
There's also some collision detection / arbitration to do, but it's definitely possible.
Most manufactured I2C devices like EEPROM or RTC's are only Slaves.
But a PIC can easily be both.
SPI can only have one Master, and it is always the Master.
SPI lines are "Driven" instead of open collector with pull-ups, and two Master's can't "Drive" the lines at the same time.
Interrupts will not affect the data transmission from a Master.
But it can ruin the response times of a Slave if not using "Clock Stretching".
Slaves must be ready and waiting to do the bidding of their Master.
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Interrupts will not affect LCDOUT either, unless your ISR is misbehaving.
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