Its perfectly fine to tow with an SUV within its ratings if that is what is best for a family. You don't need a pickup for this trailer
As to the question about suspension, it is correct that payload generally suffers. As I wrote above, its not the towing power, but the payload. By the way, my Sequoia is derived from the Toyota Tundra but differs in its use of a fully boxed frame and independent suspension (Double wishbone, coil springs, stabilizer bars IIRC).
More important than discussing the particulars of various suspension systems is the question does the TV/TT combo work within the various ratings?