You could modify Undead Fortitude. For the sake of simplicity I'm setting flat DCs, but you could factor in damage if desired. On a 15+, the zombie remains standing at 1 hp. On a 14-6, the zombie falls as though dead, but will reanimate at the start of its next turn with 1 hp unless someone successfully brain pokes it. On a 5 or less, the zombie falls dead.
Keep in mind, regarding the bite, that disease is almost trivially easy to deal with in 5e, if you have the right character in the party. A paladin can remove one disease per day for every level he has (and becomes immune to disease himself at level 3), and anyone who can cast Lesser Restoration (a 2nd level spell) can also remove disease. Unless the disease progresses very quickly, the PCs can simply wait until they're out of danger and remove it then. On the other hand, if it progresses too quickly it can become a prohibitive resource drain that forces the party to take a long rest after every zombie encounter.
In a previous campaign that I played in, the DM had the surface of the world overrun with people infected by a disease that made them rabid (in the vein of 28 Days Later). While the game took place primarily in the Underdark, we did at one point have to go to the surface to get some wyvern eggs for a wizard. We ran into quite a few of the infected in our travels, and with a good medicine check my character was able to determine which of us was infected and how long the infection would take to incubate (1d4 days). Since we could cast Lesser Restoration, we'd simply wait until the day before the character would become rabid and cure them (because if we cured them right away, they could have simply become reinfected). As such, something that was supposed to be this big, scary thing was reduced to a trivial speed bump for our adventuring party. The infected were scary, but not because they could infect us (it was because they were aggressive as hell and came at us in large numbers). Just something to consider.