For posterity's sake, here is the complete answer.
The cause
By default, the CHS uses SDP to look for all incoming serial ports, but only attempts to connect to the first one. If the port is busy - i.e. another scanner is already connected - it gives up.
The solution
Caveat emptor: This solution has not been tested on the full range of Bluetooth stacks. It should work, but there may be side-effects. If you experience any, please report them to us.
Your CHS can be configured to try to connect to up to 8 different serial ports on the host device. Instead of quitting after failing to connect to the first serial port, in Scan Mode 1
the CHS will try the next serial port until it either finds a port it can connect to or it has tried all the ports it found using SDP.
1D scanners and 8Qi
Note: If you have a laser scanner, you will need to print out these barcodes to scan them.
Factory reset your scanner
Hover to reveal barcode

Enable SPP mode
Hover to reveal barcode

Enable Scan Mode 1
Hover to reveal barcode

2D scanners (excl. 8Qi)
Factory reset your scanner
Hover to reveal barcode

Enable SPP mode
Hover to reveal barcode

Enable Scan Mode 1
Hover to reveal barcode

Revert
Don't need Scan Mode 1
anymore? Scanning the factory reset barcode for your scanner will clear all settings and put your scanner back in the default mode.