Most of those in your list that I know of are INDETERMINANT. They can get over 7 foot tall and huge. For example : Black Krim, Cherokee purple, Brandywine. So then, they need good supports (stakes, sturdy cage) and you cannot move them around. I have no experience with planting such tomatoes in containers. Lets hear from those who are experienced.
Good luck with your new house and and gardening.

I have used wrought iron plant stands like this one from Home Depot for large solid containers with 15-20 inch bases. It might work for a 20-gallon Smart Pot, although I don't know how well it would protect your lawn. I grow large indeterminate tomatoes in 20- and 25-gallon smart pots like those in my photo, including Brandywine, Cherokee purple, stump of the world, mortgage lifter, Kosovo, Earl's faux, and Omar's Lebanese.

I gave up on having a house beautiful monoculture lawn many years ago in my quest for the tastiest heirloom tomatoes. I can't grow them in the ground because my small yard is surrounded by black walnut trees that kill tomatoes. As you can probably tell, my lawn is green, but not primarily from grass. The greatest advantage from growing in fabric bags, IMO, is that it offers many of the advantages of growing in raised beds without a lot of the work. But many of those advantages depend on having the container in direct contact with the ground, which acts like a giant wick. If you're going to put fabric containers on stands that can cost as much as the containers in an effort to protect your lawn, you might decide that solid containers would be more aesthetically pleasing and easier to care for.

By the way, I used Texas Tomato Cages in my smart pots. They can be knocked over in a strong wind storm because the legs aren't sunk in the ground. That has happened only once in the past two years. But a very large tomato plant with a six-foot cage in a container on a raised plant stand might be a little more vulnerable.