Just use normal road parts for bar and stem. Aluminium is fine. Cheap is fine.
It depends on what level you expect to be racing, but I don’t think many people are concerned about their bars.
Saddle, don’t use anything to fragile - you’ll jump on and put your whole weight on it. I’ve seen carbon rails snap.
Cross-specific things… tyres!! Some consideration to wheels.
Frame. How much mud clearance does it have. How easy is it too clean - avoid lots of tiny corners and features. Have a top tube thats not digging into your shoulder.
Brakes. Hydraulic discs if you can.
All bearings should be sealed and robust.
Bar tape - this should stay grippy even if muddy and wet.
Think about how easy everything is to clean, and how much abuse it can take. Also how much will it cost to replace, because you will fall of and might damage something.