Am I correct in inferring that the new wheel has a fairly narrow rim? If not, the following may not apply.

If so the splitting you're seeing is classic for the effects of local hyper-stretching of the section between the tire's beads.

Look at this cross section diagram of a mounted tire. (Scroll down past photos until you find the drawing). Notice that the tube isn't round, but has a bulged area that fills the rim.

When you inflate a tire the tube first inflates round filling the tire, then the section spanning across the beads continues to inflate down (up in the diagram) into the rim. So the section at the rim side of the tube stretches much more than the rest. On narrow rims this difference can be very significant. If you look at your dead tube, you'll see the signs of that overstretching in the non-split areas as small bulges in places along the bottom.

The fix is to buy higher quality, stretchier tubes. Also buy the largest tube that is smaller than your tire, ie. if you have a 1.95" tires, you want a tube that says 1.95-2.25, rather than a 1.6-1.95. A larger tube won't stretch as far in the initial inflation stage, leaving more stretch for the rim.

Also rub the tubes with talc before installing, to allow the tube to slip within the tire and shift somewhat toward the rim. You might also try inflating to 5-10psi, and massaging the tire a bit on installation to help the tube equalize the uneven stretch. Both the talc and massage technique of of marginal benefit, but won't hurt. The real key is larger, higher quality tube.