TIG welding would be fine as well, although spot welds would be a lot faster! I'm sure there's some food safe braze rods and flux out there, but you'd have to research that. Out of the three options, spot welding comes out first, then TIG.
A 115V spot welder is not that expensive these days and is a good tool to have in your arsenal. Guess they have gone up to $170 at HF, they used to be around a Benjamin. I used mine recently to weld 0.050" sheet metal to 1/8" thick strapping, as well as the sheet metal to itself, when extending my lathe backsplash. The welds only take seconds. Fitting and clamping take far longer. Once clamped, you can just successively spot weld down the line, Piece of cake.