Reproductive technology is very expensive - IUI with donor sperm would only be $350-$1500, from what I can see. It looks like something like ICSI/IVF would easily be $7000-$14,500. I don't know their financial portfolio or anything, but since both parents worked and it seems HD was paying for the funeral, I don't know how much money they had available to pour into fertility treatments.
Maybe at the time he didn't mind if the baby was conceived using donor sperm, since he thought it had a chance of not working, but as time went on, it bothered him more and more. Maybe they have poor credit and didn't qualify for financing.
Just because something exists doesn't mean it's attainable for everyone. I mean, for all we know, she got sperm from a friend and used a turkey baster (the cost: a turkey baster.)
All it would take is a DNA test to prove whether or not Justin was the bio dad, any way. I don't personally think he looks a ton like dad - he has mom's coloring, mom's chin, a big smile... Even his nose looks kind of like a generic kid nose, you know? It doesn't necessarily look distinct enough, to me any way, to attribute to Justin. YMMV.