My best advice is stay low to the mat, I mean very low. Bungees are out of the question. The softer and deeper the bounce the more the mat puts the body off balance. The softer bungee rebounders are gentler on the back but the inversion and uneven bouncing can be worse on an already compromised back.
Based on nearly everyone that has bought a bungee rebounder thought it would be more gentle on their backs. It seems it would be better but in actuality it's not. For people wanting to increase bone mass I would pick a spring rebounder hands down.
If the rebounder springs don't give enough G force the body cannot stimulate much growth hormone (HGH) and the bone cells cannot rebuild. When a too soft of a rebounder has very little "pop" but instead more "throw" like a bungee cord rebounders does, the 3 g forces compared to 3 g forces on a spring are not the same. Getting 3 G forces on a bungee requires a person to jump way way high out of the mat while the spring rebounder only barely above the mat. This pop from the rebounder allows the lymphatic system to create a negative pressure in the lymph valves. Just throwing the body up really doesn't allow this. People I've talked to prefer a non jarring spring rebounder to build up their bones. Landing on your feet is how they do this and most people usually need a bar so they can use their arms to lift off to give more bounce. Those bouncing on their toes usually don't get the benefits helping their backs. If the bounce is too jarring it will be hard to land on your heels first.
My top 2 picks for people with bad backs are first, the Cellerciser. The Needak Soft bounce is my 2nd favorite for those with bad backs. With the Cellerciser you don't have to jump as high to get the same force as the needak which gives a deeper bounce. There is no inversion with the Cellerciser. Both have a super soft bounce. Based on taking both rebounders out in public and interviewing people over the years, almost everyone picks the Cellerciser and the Needak 2nd. I have the Urban, ReboundAir, Lymphaciser, Bungees, and the Lympholine in my tests. The Urban is the last one people pick and ReboundAir 2nd to the last.