Not perfect, but will work in most situations: replace the prohibited characters by something else. Since it's in keys, these new chars should be fairly rare.
/** This will replace \ with ⍀, ^$ with '₴' and dots with ⋅  to make the object compatible for mongoDB insert. 
Caveats:
    1. If you have any of ⍀, ₴ or ⋅ in your original documents, they will be converted to \$.upon decoding. 
    2. Recursive structures are always an issue. A cheap way to prevent a stack overflow is by limiting the number of levels. The default max level is 10.
 */
encodeMongoObj = function(o, level = 10) {
    var build = {}, key, newKey, value
    //if (typeof level === "undefined") level = 20     // default level if not provided
    for (key in o) {
        value = o[key]
        if (typeof value === "object") value = (level > 0) ? encodeMongoObj(value, level - 1) : null     // If this is an object, recurse if we can
        newKey = key.replace(/\\/g, '⍀').replace(/^\$/, '₴').replace(/\./g, '⋅')    // replace special chars prohibited in mongo keys
        build[newKey] = value
    }
    return build
}
/** This will decode an object encoded with the above function. We assume the structure is not recursive since it should come from Mongodb */
decodeMongoObj = function(o) {
    var build = {}, key, newKey, value
    for (key in o) {
        value = o[key]
        if (typeof value === "object") value = decodeMongoObj(value)     // If this is an object, recurse
        newKey = key.replace(/⍀/g, '\\').replace(/^₴/, '$').replace(/⋅/g, '.')    // replace special chars prohibited in mongo keys
        build[newKey] = value
    }
    return build
}
Here is a test:
var nastyObj = {
    "sub.obj" : {"$dollar\\backslash": "$\\.end$"}
}
nastyObj["$you.must.be.kidding"] = nastyObj     // make it recursive
var encoded = encodeMongoObj(nastyObj, 1)
console.log(encoded)
console.log( decodeMongoObj( encoded) )
and the results - note that the values are not modified:
{
  sub⋅obj: {
    ₴dollar⍀backslash: "$\\.end$"
  },
  ₴you⋅must⋅be⋅kidding: {
    sub⋅obj: null,
    ₴you⋅must⋅be⋅kidding: null
  }
}
[12:02:47.691] {
  "sub.obj": {
    $dollar\\backslash: "$\\.end$"
  },
  "$you.must.be.kidding": {
    "sub.obj": {},
    "$you.must.be.kidding": {}
  }
}