You could drill in pipe pile soldier beams in short, coupled sections using small micropile drill equipment. Dewater and then excavate in lifts, installing lagging and bracing as required.
Can you get a Lo-Drill into the building? If so, maybe you could dewater and install 17' minimum length soldier beams and lagging with bracing at least one level of bracing. Braces would probably interfere with the pit wall construction.
After installing some dewatering wells or well points, you might be able to dig and shore the pit with hand-installed helical soil nails and shotcrete.
You said, "It's possible the soil may be very wet." Are there any soil borings or water level readings? Don't you know where the water is? Wouldn't the water level affect the design and or performance of the new pit and equipment?
Don't forget: If you are going to form both sides of concrete pit walls, your shoring will need to be several feet wider than the concrete structure. If you intend to pour the concrete walls against the shoring (form one side of the walls), then the bracing wales will interfere with the new walls.
Don't forget to account for surcharge pressures from any nearby footings.