small welds
laser or electron beam can leave almost no bead at all. But they are expensive.
If you undercut the parts, a TIG weld can be made with little or no bead protruding. And still be strong. Micro-TIG can weld foils in the .001" range, but likely won't be strong enough.
You will have to balance the small bead with necessary strength.
Brazing may be an option. Braze can be flowed in with almost no fillet. (.015")
Good luck.
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Bill Funk
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