I wish!
I suffer from Raynaud's disease (similar to vibration white finger but it affects wider extremities such as nose and toes) and my toes are one of the things that suffer really badly on a daily basis, i've tried all sorts of so called remedies to no avail and the only thing i've found that really helps is buying boots or shoes a size larger and wearing good quality, thick socks.
If necessary in layers but not too tight that warm air from your body can't accumulate between those layers which will retain your body heat.
Socks or any clothes won't warm you up (unless artificially aided) at best they'll only retain the heat your body produces in the same way as gloves, focus instead on a 'warm core' principle and how you can firstly fuel it via your eating habits such as going out of your way to eat, for example, hot cereal oats.
Warm blood pumping to colder extremities such as finger tips and toes, fuelled by your dietary intake.
It has worked for me, it does still occasionally become borderline intolerable on a Winter riverbank but makes the difference between me getting out and don't even think about it.
Motorbike fora might be able to provide a suitable suggestion if you still want to go down the heated route, bearing in mind the complexities of charging and battery life etc.
Either way good luck, there's nothing worse than nittering cold.