You've just moved the time to write to the LCD to a different place.

When using ON INTERRUPT, it can't handle an interrupt until the current statement has finshed executing.

If you want to have a 1ms interrupt, then nothing else in the program can take more than 1ms to execute, and really should be much less than that.

If you were using ASM interrupts, then it could interrupt the LCDOUT statement, so there wouldn't be a conflict.

Or you could use Instant Interrupts to do the same thing with Basic language interrupts without the ON INTERRUPT.
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