No easy way to do this but it is possible. You need a video mixer or something that can delay the video to the monitor. You also need a good digital mixer with a good number of inputs and output that can be assigned. You can't do it with a receiver but you can do it with a preamp/processor and you insert the digital mixer between the prepro and mixer. If it is a 5.1 system you need at least 8 inputs and 6 outputs that can be assigned.

The 5.1 from the prepro goes into 6 of the inputs of the mixer and routed to 6 outputs of the mixer. Say you have 2 mics. They can be mixed into the 5 channels. you wouldn't mix it to the sub.

On the video side many video mixers have the ability to delay the HDMI video. The output of the receiver's HDMI would go into the video mixer, delay is typically needed as the digital mixer will add a little time. Output from the mixer goes to the projector.

There is more pro equipment to do this but you better have 6 figure budget.

For auditoriums and similar areas are often not surround sound and usually stereo or even mono. Then get an HDMI switcher that strips the stereo audio and add it to a mixer to go to an amp. Other than that it is probably easier and less expensive to add a separate system for the mics. Then the surround and the mic can both be used. Some companies even make an HDMI switcher with a mic input. Often used for this type of application and conference rooms.