I don't think the tarp will help with anything while its below freezing. Typically the tarps is used to increase the temperature, remove sunlight to kill the plants and if it gets hot enough to kill all weed seeds. So I doubt that will help before its below freezing. I have used it before to kill some weeds but that was late august. It won't be hot enough and will act similarity to snow.

The best method would be to do Sheet Mulching which is similar to what you are planning on doing but not 100%. I would start with a layer of cardboard, followed up by mulch / leaf litter and then finalize with compost. I would plant my plants directly in the compost and then cover in straw. You won't need the silage tarp because the cardboard weed layer should stop the grass and most of the ivy from growing. It will also start composting which the tarp will not. Around the edges the ivy can survive since the shared roots pass sugars from anything outside of the cardboard.

I have used sheet mulching to create two different small garden over the last few years. I started mine in the fall and planted in the spring which worked well. I have done it now for our second year for our vegetable garden along side some no / low dig methods. Last year was our first full year and our garden was thriving without any amendments. I expect even better results this year as soon as we can start planting.

Don't know what sillage tarp is; if it's not compostable, I wouldn't use it. I don't think just covering would kill the grass in time. Maybe it would, but it seems like if grass can survive a longer time under snow cover, it would survive under a tarp. I don't think anything would grow well just in compost and woodchips (I've heard some things do, but plants usually do need actual mineral soil). 4 inches of woodchips seems extreme for annual beds.

Could just till with a tiller or double-dig (hard work) and mix some compost in. Some grass will still probably come up, so will have to hoe every so often. After the first growing season, there'd be much less grass. And you can start using a no-till method.