very early potatoes, would this work?

I think you will need a whole lot of heat... in general plants wont grow under 7 deg C.. or very little growth.. So your 2 bags of compost wont give off enough heat.. You will possible have to have heat until end of Feb at the earliest cut off when the temps start to rise.

This is my view and may not be right.

i think i would go with plant them as deep as possible mound up and cover with as much dry straw as possible.

if in tubs then try and use insulation, as much as possible. and straw on top again.

if it's of any interest i did find 3 little potato plants in the poly tunnel today looking healthy. they wasn't planted, must've been some potatoes left from earlier in the year. not had any hard frosts yet this year though.

i put some in the poly tunnel last year before the last frost. mounded up and covered with carpet til the last of the frosts had gone. took between 6 to 8 weeks to be ready, they were ready a good month or more before the ones outside. so worth doing if you've run out of stored potato.

I tried an experiment this year which was successful in that I was eating new potatoes in mid May, about a month earlier than usual. I usually keep back some of my own 'seed' in case I need a plan B. By March they were well chitted and that was before I have taken my delivery of seed potato.
I took a 6" pot put some moist compost in it and placed the potato (Lady Christl) on top. It was put on the proverbial bedroom window sill. The potato continued to chit and it took root.
Mid April I transplanted it to a large pot. I use the supermarket cut flower black pots. Half fill with compost and top up with soil. They were placed along a south facing sheltered wall. By mid May I was eating them, when my normally planted earlies were just emerging.

I think it'll work given your location. I know that Jersey growers have over-wintered poly-tunneled crops, with the crops ready in February. If we have a frigid March like last year than probably not, but in favourable conditions I reckon you'll be eating spuds for Easter!!