Serial mice work at 1200 bps so it is correct.
MSD reports what the current setting on the chip is, not the maximum.
I am pretty confident that LapLink has some configuration setting what baud rate it should try to use, or to autodetect. You should be able to get 115200 bps. However if run MSD, it detects the serial port type as you can see. It is 16550A. I recall MSD accidentally leaves the 16550A FIFO enabled with largish timeout, so running MSD after mouse driver has loaded leads into laggy mouse cursor. If LapLink does not understand about 16550A UART and its FIFO, it is possible that LapLink also slows down because of enabled FIFO and timeout, even if LapLink uses 115200 baud rate. After cold booting the PC, do not run MSD before LapLink.
There should be no requirement for setting baud rate manually with dos MODE command.
But otherwise the parallel port allows for faster transfers. Not 4Mbit/sec though. I think I have maxed out at about 40-50 kbytes/sec with FastLynx and 4-bit parallel cable. There are also different cables that can transfer more bits at a time.
The baud rate you can achieve depends on the quality and length of the null modem cable and its configuration. FastLynx can work with 3-pin cable that only connects RXD, TXD and ground, but it can work faster when the four handshake pins are connected as well. I do not know if LapLink implements such thing.
If you transfer between a modern PC that has no real serial ports and you have to use an USB serial port, it can also be the source of sluggish transfer.
You can also use modem terminal software like Telix or Telemate or whatever to transfer files with protocols such as ZMODEM. I use this to transfer files from a Linux PC to DOS PC.