the lunar module only came out when they were by the moon to drop down

This is incorrect. Here's an Illustration of the contraption that allegedly flew the majority of the distance to the moon:

https://images.fineartamerica.com/images-medium-large-5/1-apollo-command-service-and-lunar-modules-carlos-clarivanscience-photo-library.jpg

LM is out in the open, as it would be when it allegedly descended to the moon's surface, not contained in anything.

According to NASA, the third stage engine / container thing was discarded once it had got that LM/CM contraption to a fast enough speed to get out of Earths orbit:

https://www.nasa.gov/centers/johnson/rocketpark/saturn_v_third_stage.html

NASA doesn't say exactly when that was, and instead gives a laughably broad timeline of possibility (5-112 minutes). Nor do they say at what altitude from the earth it was discarded, nor does it say it was discarded only after it left earth's orbit. (I wonder why they don't give these exact details?)

But anyway, max value 112 minutes out of a 75 hour journey time to the moons orbit is still less than 3 % of the journey. So the LM was exposed to open space for at least, and probably more than, 73 hours i.e. the majority of the journey

How does earths orbit compare with the extent of the VanAllen belts?

High earth orbit: 35,786 km

Van Allen Belts: 640 to 58,000 km.

So from NASA's own wording it seems they are saying that yes, that LM was out in the open while flying though, at the very least, about 20,000 km of the Van Allen belts, but likely more.

Just using NASAs and wikis words and stats here, not mine. I don't think that contraption went anywhere near the moon.