Hemoglobin actually isn't a direct measure of iron. You might ask them to test "Ferritin" which is an accurate measure of your son's iron stores. Also, did they check hematocrit - that's also typically used to see if (ron deficient) anemia is a problem, as well as MCV (mean corpuscle value)?
The reason hemoglobin, hematocrit nor MCV aren't necessarily accurate is because your son could actually be far more deficient than that number shows if he's also deficient in vitamin B12 (low iron tends to lower MCV and low B12 can raise it). I had this problem and so did my kids and it took forever to figure this out. I honestly don't know why the docs don't look for this, although it may be because the blood tests for B12 are not accurate - they really need to also look at folate. If that's elevated that means the body is not using the B12 that is there. 30% of the population has a genetic defect that results in having low levels of the enzymes necessary to use the B-12 and folate in the gut. This means that either sublingual, injections or transdermal B12 are necessary for any of it to bioavailable.
This is actually not an uncommon problem as they do make a supplement that has both B12 and iron, although it tastes awful. There's an excellent food based iron supplement (Floradix) that doesn't taste too bad. You can find it at Sprouts or order online from someone like www.iherb.com
There's also lots of good sublingual B vitamins (both tablets that you just dissolve (not chew) or liquids).
I just give that to my kids and have a cup of juice ready (you could try chocolate almond or rice milk).
The real reason they want him off cow's milk (and dairy in general) is that it is a very poor source of iron - and of course, if they consume alot, they certainly won't be filling up on foods high in iron. That's why all the formula's are fortified with iron (and why they tell folks not to give cow's milk as a replacement to formula prior to one year). And, if you read other posts on kids not wanting to drink milk, you can read that neither cow's milk nor dairy are necessary at all in the diet - there are plenty of other calcium based food sources. Almonds, for example, contain both iron and calcium, which is why almond milk might be a good alternative to cow's milk.