If you have a CIJ/MIJ, yes, it's Alnico. I have no idea about Squier ones.
IME, the Duncan is probably the overally fatter and smoother sounding of the bunch, taming the "small single pole piece" effect a bit.
Moreover, I don't know about Fralin's and Lollar's, but if their size specs on their sites match those on the Knuckles', then you might have the same kind of "problem" for the fit. I would not be surprised, as they all seem to echo the true vintage design with thick rope cover wound around the bobbin and push-back wires.
I can guarantee the Duncans are fine as a non-vintage replacement and fit the Japanese cavity with no issues. Not to mention that if what you are not 100% keen on is the true Tele pickup character, you'd probably feel more at home with the ¼ pounder than the Knuckles, Fralins and the likes.
I know it's a bit of a stretch, but try listening to Message In A Bottle's isolated bass track on YouTube. Recording techniques and amps apart, that to me exemplifies the rawness of the original tele single coil... Having said that, based on what you state you have all the tools in your rig to fatten and shape it to a sound close to you... But if that burp with sharp attack and relatively short sustain seems to bother you, then you probably need to stray away from the original design and the Quarter Pounder or a humbucker solution might be the way to go