Circuit todelay RS-232 TX pin closure after device power up
It sounds like you have glitches on the serial signals on power up.  This is not unusual and should be expected.

Instead of trying to delay the power coming up to the processor, its firmware should do the appropriate delays and assume the other device is in some arbitrary part of its communication protocol.  This should be normal procedure for a embedded system that communicates to remote devices over serial.

Usually the embedded system starts up its UART, waits a few 100 ms, and drains and discards all received characters.  Then it sends a sequence to put the remote device into a known state regardless of what state it might currently be in.  This might be sending a string of NULLs longer than the longest command, sending a few CR/LF in sequence, waiting some fixed time, etc.  The details are highly dependent on the byte-level and higher communication protocol.

In any case, this is a firmware problem.  Don't try to fix it with a hardware kludge.