It's unlikely that the joists are magically going to still work losing half of its cross section. You'll need to provide additional joists adjacent to the cut ones in order to replace the lost strength.

To get the right joist to span, there's no way around it we have to cut through our toilet drain pipe.
Is there any way we can divert the pipe to miss the joists?

The reverse statement of yours is just as absurd.

So adding steel framing between the wood joists that has web openings might be an option although a 6" deep framing seems very shallow and might prohibit what you need to do.

Additional support beams below these joists might be an option - but you risk losing head room clearance below.

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I have thru-bolted a steel plate to each side of the affected wood joists. Steel plate is the same depth as the joist and also has a hole to allow the pipe to run through it.

Extend the steel plate at least 2x the depth of the joist on each side of the hole. You will need to calculate the plate thickness & length, bolt diameter and pattern, etc.

Works great...…(paint the plates on the joists nearest to the toilet, if/when the toilet links the paint will help protect the plates from rusting)

Drop ceilings or horizontal chases work great too if you have the head height.

Mike McCann, PE, SE (WA)

Middle third/middle third for the hole, and when that doesn't quite make it add a steel angle above and below the hole. Not that tough if you can analyze a joist.

Very similar to a blibbit.

If the joists are 6" deep, how are you going to get the 90 degree bend in a 3" pipe to align with the center of the 6" dimension? Seems like a drop ceiling is really in order here, or defer to a new outhouse.

Mike McCann, PE, SE (WA)

Don’t forget a 3” pipe’s outside diameter is 3.5” and the pipe need to drop. Assuming 1/4” per ft and 6 joist bays @ 20”, the pipe will drop 2-1/2”

The key is reinforcement for the hole that you create. Now I don't know if what I am about to say applies for wood, nonetheless you may want to do further research on this subject. The ASME codes applying to boilers and pressure vessels construction states that holes made in pressure parts for the installation of nozzles for example need reinforcement around the holes by replacing the metal that you removed from the hole. Metal replacement for that hole would be in the form of nozzle projection into that hole and the weld metal required to secure the nozzle in that hole. Obviously there is more about reinforcement but that is the gist on this topic. MotorCity in his comment section has inadvertently touched on reinforcement by bolting steel plates on both sides of the wood joists where holes were made.

I don't think there was anything inadvertent in MotorCity's response.

"Extend the steel plate at least 2x the depth of the joist on each side of the hole. You will need to calculate the plate thickness & length, bolt diameter and pattern, etc."

I agree with this but I would add that I do not expect "2x the depth" will be nearly long enough.
Of course, I haven't calculated it for this case. ("...need to calculate..." being the key phrase).

As others have suggested, I don't think there will be enough room above and below the hole for steel angles above and below.

"horizontal chase" (aka soffit) seems like a good idea if possible.

As far as I can remember for myself, the pipe maker used metal joists for us, and of course, it depends on the ground you want to make a hole in. That was the only help I could possibly do. It is best to use metal joists for tightening as my work is done by a plumber.