Further east the better
FYI the entrance that this references is actually further west than shown on the map. Less than a mile west of the bridge over Salt River at Pebble Beach.
I love Bulldog Canyon but some spots get overrun with quads and side by sides. Fun place for four wheeling but if you want to camp I would recommend getting into Bulldog from one of the south east entrances off the 88. If you follow the trails back far enough you reach beautiful canyons and lots of sandstone and lava rock formations. 4x4 is definitely necessary if you want to go very far but I did it in a 95 Cherokee without a lift so it’s not too rough. This is pack in pack out camping, find a spot wherever you can. Make sure to get a permit in advance, you can get one in person at the ranger station off Higley day of or online if you have a few days to wait. You can definitely get to feeling like you’re in the middle of nowhere pretty quick here with the winding canyons.