ABET-accreditation is super important for some fields (like civil engineering) but not nearly as important as others. I doubt it’s overly important for construction management, but I don’t know. There are, however, 33 ABET-accredited programs in Construction Management.
Based on your son’s interests, I’d look at:
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Arizona State (behemoth but good merit aid)
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U. of New Mexico: Its Albuquerque location is definitely a city with lots of nearby outdoor opportunities, and with about 16k undergrads, it’s less than half the size of Virginia Tech. Your son would be likely to get very good merit aid here. I don’t know how they’re computing it, but it also claims a 13:1 student/faculty ratio, so class sizes might be much more approachable.
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North Carolina A&T: I’ve heard some great things about this HBCU, especially for your son’s interests. Sticker price will bring you to Virginia Tech costs, and merit aid may be available as well. Not as accessible to nature areas as Appalachian State, but not far.
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UNC – Charlotte: Tons of construction and growth in the Charlotte area, and similarly to A&T, not too far from nature areas.
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Norwich (VT): This school has about 3k undergrads and will have lots of small classes. And Vermont has stellar outdoors opportunities.
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Western Carolina, already mentioned upthread, is also ABET-accredited.
There are 82 schools that have construction management or similar as a Bachelor’s degree.
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Bradley (IL): Not exactly an outdoors mecca, but it seems like a school that might be a good fit. There are about 4300 undergrads here.
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John Brown (AR): This is a university where my sense is that religion plays an important role. If that works for your son, this school with about 1800 undergrads would provide lots of attention for him and he’d be near the Ozarks.
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Ohio Northern: Again, not exactly the ideal place for an outdoors enthusiast, but I think your son would get lots of attention here and I’ve heard a number of people speaking positively about this one. About 2400 undergrads here.
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Roger Williams (RI): This school of about 4200 undergrads is right by the water for sailing and other activities.
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SUNY Environmental Science & Forestry: This school is small and niche (about 1600 undergrads), but students can also take classes at neighboring Syracuse and participate in its clubs and activities. With an 11:1 student/faculty ratio, your son should get lots of individualized attention.
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U. of Denver (CO): If you think your son is going to fall in love with Colorado, then your family may want to investigate this school. I don’t know if DU mis-categorized, but it listed 0 construction management majors in the most recent graduating class, but this site describes the major it offers. There are about 5900 undergrads here.
If your son’s looking for other big schools that offer the major, then U. of Tennessee has the close proximity to home and mountains, but there’s also Boise State and Northern Arizona if he wants to go further afield.