I see. My bass seems to be good with fundamentals, however, I tend to play it direct and with in ear monitors. My IEMs are high passed at 60hz and I let the rumble from FOH provide the oomph. When it is rarely played at home through an amp, it seems fine-nothing seems to be missing, compared to any other bass I play at home.
I wouldn't be surprised if the preamp of the Squire is not the same tuning as the MIM/US pre, regarding the high and low pass filtering, frequency centers, etc... The quality of Squier preamps is certainly not of the same build and tonal quality of the domestic Fenders, have seen that many times before.
Another thing with single H Dimensions (as well as Stingrays) is that pickup location. As the pickup gets closer to the bridge, the harmonic structure changes a lot. I have a Peavey Patriot with an added Super Ferrite single coil neck pickup (centered right where the 24th fret would be). That pickup soloed has a deep and solid fundamental, moreso than the same pickup in the P sweet spot (like on my Foundation) or the bridge pickup of the Patriot (which is pretty much where a Stingray pickup resides--which is punchy, but not a solid fundamental tone.