To map LocalTime as a keyword type (which I think is what you want, which will not be analyzed, but still be indexed and searchable), you can use
public class MyItem
{
[JsonProperty("start")]
[JsonConverter(typeof(LocalTimeConverter))]
[Keyword]
public LocalTime Start { get; set; }
}
And create the index and mapping as you are currently doing.
You could omit the JsonPropertyAttribute too if you wanted as NEST camel cases property names by default.
This produces the mapping
{
"mappings": {
"myitem": {
"properties": {
"start": {
"type": "keyword"
}
}
}
}
}