Winter weather can seem insurmountable to Seattle-ites and southerners but millions of people survive winter in the middle of the country every year. And tens of thousands more survive travelling in mid-winter too. (though most of those wear snow boots rather than Tevas and wool socks......)
If you haven't driven on snow that's a skill that needs to be picked up. Taking some time to slide around a snowy parking lot along the way on the trip would be helpful. I'd do that with my UHaul truck if it were me as every vehicle handles differently and you need to know what to expect. (I'd also demand that it have good deep tire tread before accepting it.)
You're a mom - it's your job to worry. But I wouldn't expect to talk a manly shipyard welder out of the manly adventure of moving cross-country in the winter. It's not a suicide mission - it's just a challenge and maybe only an inconvenience.