Heavy blanks can be a challenge to get mounted singlehandedly, but I've discovered a couple tricks that make it easier. I have a couple choices of drive options available: An Elio drive and a big spur drive that fits into my 50mm chuck jaws. I have no recollection of who made the spur drive or where I got it, but it tends to get more use because the Elio requires use of an Allen key to bury the pointed machine screws. (I keep thinking I should replace them with pointed flange bolts and keep the proper size Gear Wrench handy)
For either use of either drive, I start by choosing a center point. Big blanks are too heavy to try to find a balance point, so it's estimated, or if the size of the piece dictates it, the center point is dictated by measurement. It can be on the cut face or the bark side, you just have to start somewhere. It can always be changed/adjusted later. I use a Forstner bit to drill into the blank a bit. This not only provides a solid grip point, but provides a shoulder to sort of "hang" the blank on the drive point, while the other hand brings up the tailstock.
As far as your piece not running true when you flip it around, are you making a random size tenon? I used to see chuck jaws as having a range of allowable sizes, which is true, but to flip a piece and have it run true, (or very close) the tenon has to be cut to a size that is very close to being too small. (chuck jaws nearly closed up completely) Larger sized tenons depend on the corners of the jaws, and varying density of the wood, they bite in unevenly and grip it off-center.
I use a template to accurately mark the tenon size. I cut close to the line, creating a rough tenon. I continue shaping and refining the bowl from the bottom to the top of rim. Before I remove it, I make the final cuts on the tenon. With a sharp spindle gouge, I cut the line with a slight dovetail angle, and cut a flat shoulder flat to meet it. Usually when I flip a piece, I tighten the jaws just enough to hold it, then I bring up the tailstock (with a scrap between) to provide some pressure. I lock the spindle and spin the blank in the chuck just slightly to make sure there is good solid contact between the shoulder and jaw faces, and then tighten the chuck.