My fuller (tested and stable) solution is to have the convenient alt+ctrl[+shift]+k behave like [shift+]f3 in all contexts as expected:
  //--------------------------------------------------------------------
  // __For_nested_snippets
  //--------------------------------------------------------------------
{
    "keys": [
      "alt+ctrl+k"
    ],
    "command": "next_field",
    "context": [
      {
        "key": "has_next_field",
        "operator": "equal",
        "operand": true
      }
    ]
  },
  {
    "keys": [
      "alt+ctrl+shift+k"
    ],
    "command": "prev_field",
    "context": [
      {
        "key": "has_next_field",
        "operator": "equal",
        "operand": true
      }
    ]
  },
  {
    "keys": [
      "tab"
    ],
    "command": "insert_best_completion",
    "args": {
      "default": "\t",
      "exact": false
    },
    "context": [
      {
        "key": "has_next_field",
        "operator": "equal",
        "operand": true
      }
    ]
  },
    //--------------------------------------------------------------------
  // __Find_Next
  //--------------------------------------------------------------------
{
    "keys": [
      "alt+ctrl+k"
    ],
    "command": "find_next"
  },
  {
    "keys": [
      "alt+ctrl+shift+k"
    ],
    "command": "find_prev"
  },
    //--------------------------------------------------------------------
  // __Latex_commands
  //--------------------------------------------------------------------
  {
    "keys": [
      "shift+alt+ctrl+k"
    ],
    "command": "prev_bookmark",
    "context": [
      {
        "key": "selector",
        "operator": "equal",
        "operand": "text.tex.latex"
      }
    ]
  },
  {
    "keys": [
      "alt+ctrl+k"
    ],
    "command": "next_bookmark",
    "context": [
      {
        "key": "selector",
        "operator": "equal",
        "operand": "text.tex.latex"
      }
    ]
  },