I am aware that Silicon Labs provides their Bluetooth products with a stack that is capable of multiple connections of up to 8 sinks. Not all module vendors may offer this, but since you make reference to an SL device it may be worth knowing there is support for what you are after. However the code implementation may be a bit more tricky ...
Here is a brief summary of what occurs after pairing with all the possible sink devices (up to 8).
As you may or may not already know, in the configuration you are interested in, Bluetooth connections use small connection windows to exchange data in every connection interval. At least an empty packet is sent in every connection interval, first by the master, then by the slave, and if there is data to send, then the packets are extended. This ensures that no continuous receiving is needed, and in most of the time the device can go into sleep mode. The same behavior makes it possible to keep alive multiple connections easily at the same time. The connection windows of different connections are simply interleaved.
The article in question is KBA_BT_0907 and you may find it usefull. Good luck !