On some ports on RTL872x devices you may also need to add a short delay, because flush writes out the output buffer, but there's a second byte FIFO that still needs to be output. It only applies to one of the alternate ports, not the regular Serial1, if I recall correctly.
Though I would use a SC16IS7xx SPI or I2C UART instead of using a multiplexer. The SC16IS752 dual port works great with all Particle devices using the SC16IS7xxRK library. I'd use SPI mode for 115200.