I personally wouldn't want my kids riding in an aftermarket seat on a daily basis. Perhaps for a single trip if absolutely necessary, provided the seatbelt fit alright. Probably can't be any worse that letting a kid ride in a classic car or other less than ideal arrangement. On the other hand, with no standards, who knows if it is anywhere near as safe as a factory third seat.
I have to wonder what "DOT Approved" means. I'd guess there is no federal safety standard that applies to aftermarket seating systems. I find it very hard to believe there is any money for standards or government mandateds and monitored testing to be designed and implemented for such a small market.
Auto companies do a huge amount of internal design, simulations and crash testing on each model they sell, even for crash types not covered by standard crash testing requirements (like rear enders). There's a lot of liability involved if they ever have another Ford Pinto fiasco... I'd also guess that a small aftermarket seat company doesn't have remotely enough engineering or testing capability to do this for their product in a single model of automobile, let alone in each and every one for which they sell their product. The costs would be staggering to do this for a product with a relatively small price and small unit sales.
I may be wrong, but I wouldn't bet my kids' life on it.