You will add quite a few miles just driving around the NP's and City's, but even so your trip is going to be quite easy to achieve with a month. Remember also that drive times that come from a computer are wildly optimistic when it comes to 'real world' driving. Again, your journeys are all perfectly doable just allow longer, 20% longer is normally a safe bet.

With a stopover from LA you could possibly visit Joshua Tree NP and on the way to Santa Fe, you could possibly include Monument valley and Four corners.

It would be possible to visit both Bryce and Zion NP and I would recommend doing so. You could break the journey with an overnight in Moab and visit Arches or Canyonlands NP and then take Utah scenic 12 to Bryce canyon.

Bishop is not an ideal place to stay on route to Yosemite, simply because you will not be able to cross the Sierra Nevada on CA120 [Tioga Pass] when you travel. It will be closed due to winter snow accumulation and you will have to go south around the mountains. Lake Isabella would be an option but you could push a little further. With travel times I would actually recommend another night in Yosemite, or close by.

You won't need a Jeep/SUV for your trip. An ordinary mid-size Sedan would do the job just fine as will a Mustang if it has room for all your gear, although usualy you only rent a 'class' of vehicle rather than a specific make and model so you would have to check that out.