this is english walnut, so low juglone. The root system will be an asset to your garden, once it is well dead and decomposing. A decent way to kill the tree is to cut it flush with the soil, then paint Round Up around the cambium.
I would even suggest that you bury whatever branches are left, or save the wood chips for mulching in future years. Build the beds on top of the stumps. The juglone will disappear quickly once they are dead.

You could also buy a wood stove with catalytic converter, and use the wood ash to fertilize your beds (only add nitrogen). Walnut wood is high quality, probably there will be someone willing to pay for it.