Bachelor-level degrees are sometimes referred to as "undergraduate degrees". The expectation is that the graduate will go on to do postgraduate degrees.

Since the qualification is bachelor of [pick your subject], then there is no possessive involved - the "bachelor" is merely an adjective describing "degree". Therefore it is "bachelor degree", although I have to agree with FS about its problems. The use of "undergraduate degree" probably evolved, at least in part, because of this awkwardness.