The SD DMV is extremely easy to deal with (Clay county being the recommended office). Everything is done by mail. No vin inspections when purchasing a new vehicle, no emissions, etc. I've bought/sold SD titled vehicles that have never been in the state.
Insurance is a little grey, but no more than any full timer who really has no primary garaged address.. Most SD residents use their mail forwarding address as the "garaged address" which you could as well. You will want SD insurance, regardless of your DL state.
I'm sure all states say their residents must register their vehicles in their state, but that is because 99.9% of people live/work & own property in the same state. One state generally has no jurisdiction to impose taxes/fees on property with out of state nexus. For example, TX can't impose property tax on an out of state rental property owned by a TX resident.
SD will have no issues if you are stopped in SD with SD tagged tags and a TX drivers license. Read the link I posted earlier. SD has issued multiple memos on the subject of out of state resident registration. Carry a copy of said memos in your glovebox if necessary.
I would be more "nervous" about driving around TX with a TX DL in an SD tagged vehicle in your name. This is a common scenario in the NW, where WA residents will purchase/register their cars in OR due to having no sales tax and extremely cheap registration. WA has cracked down on this, but it is obviously a tax evasion scenario because OR residents garaging their cars in OR, should register their cars in OR. This senario is a bit different though, since you would be a TX resident with an Out of state vehicle that is only temporarily in-state. This whole problem revolves around TX residents who are very rarely in TX, so this really shouldn't be a large concern.
I would like to see the affidavit if anyone has a copy of it. They really have no jurisdiction over out of state property, and I bet the TX DMV would agree (once you found the right person to talk to there).