Yes, it is possible, in theory at least and with some customisation of parts.
But; Bluetooth audio has a delay due to buffering etc. which means anything you hear through it would be slightly out of step with anything you heard directly, or peoples mouth movements etc.
And your own voice, which tends to make people stutter if that is delayed.
Some form of analog wireless or non-bluetooth digital system would be better as there is no noticeable delay.
You can buy professional gear for stage use, which it may be possible to configure in that manner, but it would be expensive.
A wired system would be a fraction the price.
The only relatively low cost off-the-shelf style of device I can think of is a wireless Karaoke mic that has a receiver with headphone monitoring, if there is such a thing?