All VDOT maps and documents refer to them as state highways, which is how they were designated by the CTB. They have never been submitted to AASHTO (nor would 258 ALT make sense to be as there is a 258 Bus and Byp there).
How good is VDOT about getting AASHTO approval for auxiliary US routes? Some states seem not to bother.
I, too, would leave them as auxiliary US routes as they're posted and in the HB, even if they technically shouldn't have been. Unless you're convinced that the US route signage is just an isolated sign-o, like the one I've seen on VA 311.
Had I drafted the set I would've included 341 and 357 as they can be done in two segments. 314 and 319 have multiple segments although i could probably create a logical path that encompasses a fair representation of those routes. I gave him the posting status and number of segments for all the 3xx routes and he chose the ones posted in a full shield and were 1 or 2 segments.
How many segments of those routes (all of which are within state facilities, or correcting them to other primary highways) are posted with standard route markers (not just smaller white rectangles)? ISTM, from your site, that all those routes have at least one fully-posted segment, even if the branches technically assigned to those routes aren't. Normal practice is to include signed segments no matter how many, and ignore the unsigned ones (for example, MD 800). That seems to have been done for VA 321, which seems to include multiple roads on the College of William & Mary campus, but the draft HB has only the one segment starting off-campus. For an example where branches were included, look at UT 282 (main route looping through the U. of Utah campus, with two branch routes in separate route files).
I would've also done the 1 or 2 segment 3xx routes that are posted in white rectangles because Virginia uses those for all kinds of primary routes including US routes.
FWIW, Hawaii has some state routes, like HI 901 and HI 5600, with number signage limited to
small number plates under milemarkers (smaller than the Virginia white rectangles). Tim told me to treat them all as unsigned. I'm disinclined to change that, because the signed/unsigned (or barely signed) distinction helps weed out minor routes in a state like Hawaii, Maryland, or (apparently) Virginia that feels compelled to slap a route number on every road it maintains.
EDIT: On a completely separate topic, what about adding VA 228 Truck in Herndon? Your website notes a well-posted VA 228 Truck, which loops around downtown Herndon, onto which the town would like VA 228 rerouted.